Section Limit.
Variable A : Set.
Variable eq : A -> A -> Prop.
Hypothesis eq_dec : forall (a1 a2 : A),
{eq a1 a2} + {~eq a1 a2}.
Hypothesis eq_refl : forall (a : A),
eq a a.
Hypothesis eq_symm : forall a1 a2,
eq a1 a2 ->
eq a2 a1.
Variable distance : A -> A -> A.
Hypothesis distance_comm : forall a1 a2,
eq (distance a1 a2) (distance a2 a1).
Variable half : A -> A.
Variable zero : A.
Hypothesis half_nz : forall (a : A),
~eq a zero ->
~eq (half a) zero.
Hypothesis distance_eq_zero : forall (a1 a2 : A),
eq a1 a2 <->
eq (distance a1 a2) zero.
Hypothesis distance_eq_compat : forall a1 a2 a3 a4,
eq a1 a3 ->
eq a2 a4 ->
eq (distance a1 a2) (distance a3 a4).
Variable le : A -> A -> Prop.
Hypothesis le_eq : forall (a1 a2 : A),
eq a1 a2 ->
le a1 a2.
Hypothesis le_trans : forall (a1 a2 a3 : A),
le a1 a2 ->
le a2 a3 ->
le a1 a3.
(* all numbers positive *)
Hypothesis all_pos : forall (a : A),
le zero a.
Hypothesis le_impl_eq : forall (a1 a2 : A),
le a1 a2 ->
le a2 a1 ->
eq a1 a2.
Variable plus : A -> A -> A.
Variable plus_0_eq : forall (a1 a2 : A),
eq a2 zero ->
eq a1 (plus a1 a2).
Variable plus_le_compat : forall a1 a2 a3 a4,
le a1 a2 ->
le a3 a4 ->
le (plus a1 a3) (plus a2 a4).
Hypothesis triangle_inequality : forall (a1 a2 a3 : A),
le (distance a1 a2) (plus (distance a1 a3) (distance a3 a2)).
Hypothesis half_plus : forall (a : A),
eq (plus (half a) (half a)) a.
Hypothesis le_half : forall (a : A),
le a (half a) ->
eq a zero.
intuition.
destruct (eq_dec a zero).
trivial.
exfalso.
specialize (H (half a)).
apply f.
eapply le_half.
apply H.
eapply half_nz.
trivial.
Qed.
Definition left_total(r : nat -> A -> Prop) :=
forall n, exists a, r n a.
Definition functional(r : nat -> A -> Prop) :=
forall n a1 a2,
r n a1 ->
r n a2 ->
eq a1 a2.
Definition inf_limit(f : nat -> A -> Prop)(a : A) :=
forall (epsilon : A),
~eq epsilon zero ->
exists n : nat,
forall (n' : nat),
n' >= n ->
forall a',
f n' a' ->
le (distance a' a) epsilon.
(* Note: if the codomain of the relation is empty (after any point), then we can assign any limit to it. *)
(* We could define inf_limit in terms of inf_limit_2, but inf_limit_2 only shows up in proofs, while inf_limit shows up in top-level definitions. *)
Definition inf_limit_2(f f' : nat -> A -> Prop) :=
forall (epsilon : A),
~eq epsilon zero ->
exists n : nat,
forall (n' : nat),
n' >= n ->
forall a a',
f n' a ->
f' n' a' ->
le (distance a a') epsilon.
intuition.
Abort.
(*  remember half as h. *)
intuition.
unfold inf_limit in *.
intuition.
specialize (H1 (half epsilon)).
specialize (H2 (half epsilon)).
destruct H1.
intuition.
eapply half_nz; eauto.
destruct H2.
intuition.
eapply half_nz; eauto.
destruct (H (max x x0)).
specialize (H1 (max x x0)).
specialize (H2 (max x x0)).
eapply le_trans.
eapply (triangle_inequality _ _ x1).
eapply le_trans.
eapply plus_le_compat.
eapply le_trans.
eapply le_eq.
eapply distance_comm.
eapply H1.
eapply Max.max_lub_l; eauto.
trivial.
eapply H2.
eapply Max.max_lub_r; eauto.
trivial.
eapply le_eq.
eapply half_plus.
Qed.
intuition.
eapply distance_eq_zero.
eapply le_epsilon_zero.
intuition.
eapply limit_eq_h; eauto.
Qed.
unfold inf_limit; intuition.
destruct (H epsilon).
trivial.
exists x.
intuition.
eapply H2.
eapply H3.
eapply H0.
trivial.
Qed.
End Limit.
(* limits for Rat *)
Local Open Scope rat_scope.
Definition rat_inf_limit(f : nat -> Rat -> Prop)(r : Rat) :=
inf_limit eqRat ratDistance rat0 leRat f r.
Definition rat_inf_limit_2(f1 f2 : nat -> Rat -> Prop) :=
inf_limit_2 eqRat ratDistance rat0 leRat f1 f2.
Definition rat_limits_eq :=
limits_eq eq_Rat_dec ratDistance_comm ratHalf ratHalf_ne_0 ratIdentityIndiscernables eqRat_impl_leRat leRat_trans ratAdd ratAdd_leRat_compat ratTriangleInequality ratHalf_add le_ratHalf_0.
unfold rat_inf_limit_2, inf_limit_2; intuition.
edestruct (H (ratHalf epsilon)).
eapply ratHalf_ne_0; intuition.
edestruct (H0 (ratHalf epsilon)); eauto.
eapply ratHalf_ne_0.
intuition.
exists (max x x0).
intuition.
destruct (H1 n').
eapply leRat_trans.
eapply ratDistance_le_trans.
eapply H3.
eapply Max.max_lub_l.
eapply H5.
trivial.
eapply H8.
eapply H4.
eapply Max.max_lub_r.
eapply H5.
eauto.
trivial.
eapply eqRat_impl_leRat.
eapply ratHalf_add.
Qed.
unfold rat_inf_limit, inf_limit in *.
intuition.
destruct (H epsilon); intuition.
destruct (H0 epsilon); intuition.
exists (max c (max x x0)).
intuition.
destruct (H3 n').
destruct (H4 n').
eapply leRat_trans.
eapply ratDistance_le_max.
eapply H1.
eapply Max.max_lub_l.
eauto.
eapply H9.
eapply H10.
eapply H2.
eapply Max.max_lub_l.
eauto.
eapply H9.
eapply H11.
eapply maxRat_leRat_same; eauto using Max.max_lub_l, Max.max_lub_r.
Qed.
unfold rat_inf_limit, inf_limit.
intuition.
destruct (H epsilon); intuition.
econstructor.
intuition.
specialize (H1 (div2 n')).
eapply H1.
eapply div2_ge; eauto.
eauto.
Qed.
unfold rat_inf_limit, inf_limit in *.
intuition.
edestruct (H1 (ratHalf epsilon)).
eapply ratHalf_ne_0.
intuition.
edestruct (H2 (ratHalf epsilon)).
eapply ratHalf_ne_0.
intuition.
exists (Max.max x x0).
intuition.
unfold ratAdd_rel in *.
destruct (H n').
destruct (H0 n').
rewrite H7; eauto.
eapply leRat_trans.
eapply rat_distance_of_sum.
rewrite H4.
rewrite H5.
rewrite ratHalf_add.
intuition.
eapply Max.max_lub_r.
eauto.
trivial.
eapply Max.max_lub_l.
eauto.
trivial.
Qed.
unfold rat_inf_limit, inf_limit in *.
intuition.
destruct (le_Rat_dec c1 c2).
edestruct (H1 (ratHalf epsilon)).
eapply ratHalf_ne_0.
intuition.
edestruct (H2 (ratHalf epsilon)).
eapply ratHalf_ne_0.
intuition.
exists (Max.max x x0).
intuition.
unfold ratSubtract_rel in *.
destruct (H n').
destruct (H0 n').
rewrite H7; eauto.
rewrite (ratSubtract_0 l).
eapply ratDistance_0_r_le.
eapply leRat_trans.
eapply ratSubtract_partition_leRat.
eapply ratSubtract_ratDistance_le.
2:{
rewrite H4.
rewrite ratHalf_add.
intuition.
eapply Max.max_lub_l.
eauto.
trivial.
}
eapply leRat_trans.
eapply ratSubtract_partition_leRat.
rewrite ratSubtract_0.
eapply leRat_refl.
eapply l.
eapply ratSubtract_ratDistance_le.
rewrite ratDistance_comm.
rewrite H5.
rewrite <- ratAdd_0_l.
intuition.
eapply Max.max_lub_r.
eauto.
trivial.
edestruct (H1 (minRat (ratHalf epsilon) (ratHalf (ratSubtract c1 c2)))).
intuition.
unfold minRat in *.
destruct (bleRat (ratHalf epsilon) (ratHalf (ratSubtract c1 c2))).
eapply ratHalf_ne_0 in H4;
intuition.
eapply ratHalf_ne_0 in H4;
intuition.
eapply n.
eapply ratSubtract_0_inv.
trivial.
edestruct (H2 (minRat (ratHalf epsilon) (ratHalf (ratSubtract c1 c2)))).
intuition.
unfold minRat in *.
destruct (bleRat (ratHalf epsilon) (ratHalf (ratSubtract c1 c2))).
eapply ratHalf_ne_0 in H5;
intuition.
eapply ratHalf_ne_0 in H5;
intuition.
eapply n.
eapply ratSubtract_0_inv.
trivial.
exists (Max.max x x0).
intuition.
destruct (H n').
destruct (H0 n').
unfold ratSubtract_rel in *.
rewrite H7; eauto.
assert (x2 <= x1).
eapply leRat_trans.
eapply ratDistance_le_sum.
eapply H5.
eapply Max.max_lub_r.
eauto.
trivial.
eapply leRat_trans.
eapply ratAdd_leRat_compat.
eapply leRat_refl.
eapply minRat_le_r.
eapply (leRat_ratAdd_same_r (ratHalf (ratSubtract c1 c2))).
rewrite ratAdd_assoc.
rewrite ratHalf_add.
rewrite ratSubtract_ratAdd_inverse_2.
eapply ratDistance_le_sum.
eapply leRat_trans.
rewrite ratDistance_comm.
eapply H4.
eapply Max.max_lub_l.
eauto.
trivial.
eapply minRat_le_r.
case_eq (bleRat c1 c2); intuition.
eapply bleRat_total.
trivial.
eapply leRat_trans.
eapply rat_distance_of_difference.
trivial.
case_eq (bleRat c1 c2); intuition.
eapply bleRat_total.
trivial.
eapply H4.
eapply Max.max_lub_l.
eauto.
trivial.
eapply H5.
eapply Max.max_lub_r.
eauto.
trivial.
rewrite minRat_le_l.
rewrite ratHalf_add.
intuition.
Qed.
(* This is the proof found at http://planetmath.org/ProofOfLimitRuleOfProduct.html *)
unfold rat_inf_limit, inf_limit in *.
intuition.
edestruct (H1 ((ratInverse (1 + c2)) * epsilon * (1/ 2))).
intuition.
apply ratMult_0 in H4.
intuition.
apply ratMult_0 in H5.
intuition.
eapply ratInverse_nz.
eauto.
edestruct (H2 ((ratInverse (1 + c1)) * epsilon * (1 / 2))).
intuition.
apply ratMult_0 in H5.
intuition.
apply ratMult_0 in H6.
intuition.
eapply ratInverse_nz.
eauto.
edestruct (H2 1).
intuition.
exists (maxList (x :: x0 :: x1 :: nil)).
intuition.
unfold ratMult_rel in *.
edestruct H.
edestruct H0.
rewrite H8; eauto.
eapply leRat_trans.
eapply (ratTriangleInequality _ _ (c1 * x3)).
rewrite ratMult_ratDistance_factor_r.
rewrite ratMult_ratDistance_factor_l.
rewrite ratMult_comm.
eapply leRat_trans.
eapply ratAdd_leRat_compat.
eapply ratMult_leRat_compat.
eapply ratDistance_le_sum.
eapply H6; [idtac | eapply H10].
eapply le_trans.
eapply (maxList_correct (x :: x0 :: x1 :: nil)).
simpl.
intuition.
trivial.
eapply H4; [idtac | eapply H9].
eapply le_trans.
eapply (maxList_correct (x :: x0 :: x1 :: nil)).
simpl.
intuition.
trivial.
eapply ratMult_leRat_compat.
eapply (ratAdd_any_leRat_l 1).
eapply leRat_refl.
eapply H5; [idtac | eapply H10].
eapply le_trans.
eapply (maxList_correct (x :: x0 :: x1 :: nil)).
simpl.
intuition.
trivial.
eapply eqRat_impl_leRat.
repeat rewrite <- ratMult_assoc.
rewrite (ratAdd_comm c2).
rewrite (ratAdd_comm c1).
rewrite (ratMult_assoc ((1 + c2) * (ratInverse (1 + c2)))).
rewrite (ratMult_assoc ((1 + c1) * (ratInverse (1 + c1)))).
rewrite <- ratMult_distrib_r.
rewrite (ratMult_comm (1 + c2)).
rewrite (ratMult_comm (1 + c1)).
repeat rewrite ratInverse_prod_1.
rewrite ratMult_2.
rewrite ratMult_1_l.
rewrite ratMult_comm.
rewrite ratMult_assoc.
rewrite ratMult_eq_rat1.
eapply ratMult_1_r.
intuition.
assert (1 <= 0).
eapply leRat_trans.
eapply (ratAdd_any_leRat_r c1).
eapply leRat_refl.
eapply eqRat_impl_leRat.
rewrite ratAdd_comm.
trivial.
intuition.
intuition.
assert (1 <= 0).
eapply leRat_trans.
eapply (ratAdd_any_leRat_r c2).
eapply leRat_refl.
eapply eqRat_impl_leRat.
rewrite ratAdd_comm.
trivial.
intuition.
Qed.
Definition rat_limit(f : Rat -> Rat -> Prop)(p L : Rat) :=
forall epsilon,
~ (epsilon == 0) ->
exists delta, (~delta == 0) /\
forall x v,
f x v ->
(ratDistance x p) <= delta ->
(ratDistance v L) <= epsilon.
Definition continuous_at(f : Rat -> Rat -> Prop)(c : Rat) :=
forall v, f c v ->
rat_limit f c v.
unfold continuous_at, rat_limit, rat_inf_limit, inf_limit in *.
intuition.
edestruct H1; eauto.
intuition.
destruct (H x).
trivial.
exists x0.
intuition.
destruct (H2 n').
eapply H6.
eauto.
eapply H4.
eapply H7.
trivial.
Qed.
intuition.
unfold rat_inf_limit_2, inf_limit_2, rat_inf_limit, inf_limit in *.
intuition.
edestruct (H (ratHalf epsilon)).
eapply ratHalf_ne_0.
intuition.
edestruct (H0 (ratHalf epsilon)).
eapply ratHalf_ne_0.
intuition.
exists (Max.max x x0).
intuition.
destruct (H1 n').
eapply leRat_trans.
eapply ratTriangleInequality.
rewrite H3.
rewrite H4.
rewrite ratHalf_add.
intuition.
eapply Max.max_lub_r.
eauto.
eauto.
eapply Max.max_lub_l.
eauto.
trivial.
trivial.
Qed.
unfold rat_inf_limit, inf_limit in *.
intuition.
edestruct H; eauto.
exists x.
intuition.
rewrite <- H0.
eapply H2.
eapply H3.
trivial.
Qed.
unfold rat_inf_limit, inf_limit in *.
intuition.
exists O.
intuition.
eapply leRat_trans.
eapply eqRat_impl_leRat.
rewrite <- ratIdentityIndiscernables.
eauto.
eapply rat0_le_all.
Qed.
induction ls; intuition.
unfold rat_inf_limit, inf_limit.
intuition.
exists O.
intuition.
inversion H4; clear H4; subst.
unfold sumList.
simpl.
rewrite H5.
eapply leRat_trans.
eapply eqRat_impl_leRat.
rewrite <- ratIdentityIndiscernables.
intuition.
eapply rat0_le_all.
eapply (@rat_inf_limit_trans _ (fun n : nat => ratAdd_rel (f1 a n) (sumList_rel (fun a0 : A => f1 a0 n) ls))).
unfold rat_inf_limit_2, inf_limit_2.
intuition.
exists O.
intuition.
inversion H5; clear H5; subst.
unfold ratAdd_rel in *.
rewrite H6.
rewrite H12.
eapply leRat_trans.
eapply eqRat_impl_leRat.
rewrite <- ratIdentityIndiscernables.
eapply eqRat_refl.
eapply rat0_le_all.
trivial.
trivial.
eapply rat_inf_limit_eq.
eapply rat_inf_limit_sum.
eauto.
unfold left_total; intuition.
eapply sumList_rel_left_total.
intuition.
eapply H0.
eapply H.
eauto.
unfold sumList.
simpl.
symmetry.
rewrite fold_add_body_eq; [idtac | eapply ratAdd_comm | idtac].
rewrite fold_add_init.
eapply eqRat_refl.
intuition.
unfold left_total; intuition.
eapply ratAdd_rel_left_total.
eapply H0.
eapply sumList_rel_left_total.
intuition.
eapply H0.
eauto.
intuition.
eapply sumList_rel_func; eauto.
intuition.
eauto.
Qed.
unfold continuous_at, rat_limit in *.
intuition.
exists (minRat (c * (1 / 2)) (c * (1/2) * c * epsilon)).
intuition.
unfold minRat in *.
destruct (bleRat (c * (1 / 2)) (c * (1 / 2) * c * epsilon)).
apply ratMult_0 in H2; intuition.
apply ratMult_0 in H2; intuition.
apply ratMult_0 in H3; intuition.
apply ratMult_0 in H2; intuition.
rewrite H2.
rewrite H0.
assert (ratDistance x c <= c * (1/2)).
eapply leRat_trans.
eapply H3.
eapply minRat_le_l.
assert (ratSubtract c (c * (1/2)) <= x).
eapply ratDistance_ge_difference.
rewrite ratDistance_comm.
trivial.
assert (x <= c + (c * (1/2))).
eapply ratDistance_le_sum.
trivial.
assert (c * (1/2) <= x).
rewrite <- (ratSubtract_half).
trivial.
clear H5.
assert (x <= c * (3/2)).
rewrite ratMult_ratAdd_cd in H6.
simpl in *.
trivial.
clear H6.
assert (ratInverse x <= ratInverse (c * (1 / 2))).
apply ratInverse_leRat.
intuition.
apply ratMult_0 in H6; intuition.
trivial.
rewrite ratDistance_ratInverse.
rewrite H3.
rewrite minRat_le_r.
rewrite ratInverse_ratMult; intuition.
rewrite H6.
rewrite ratInverse_ratMult; intuition.
rewrite ratMult_comm.
repeat rewrite <- ratMult_assoc.
rewrite (ratMult_assoc ((ratInverse c) * (ratInverse (1/2)))).
rewrite ratInverse_prod_1; intuition.
rewrite ratMult_1_r.
rewrite (ratMult_assoc (ratInverse c)).
rewrite ratInverse_prod_1; intuition.
rewrite ratMult_1_r.
rewrite ratInverse_prod_1; intuition.
rewrite ratMult_1_l.
intuition.
rewrite H8 in H7.
assert (c * (1 / 2) == 0).
eapply leRat_impl_eqRat.
trivial.
eapply rat0_le_all.
apply ratMult_0 in H9; intuition.
intuition.
rewrite H8 in H7.
assert (c * (1 / 2) == 0).
eapply leRat_impl_eqRat.
trivial.
eapply rat0_le_all.
apply ratMult_0 in H9; intuition.
intuition.
Qed.
intuition.
eapply rat_inf_limit_comp.
eauto.
symmetry.
trivial.
eapply ratInverse_continuous.
trivial.
trivial.
Qed.
intuition.
unfold rat_inf_limit, inf_limit in *.
intuition.
edestruct (H (1 / 2 * (ratSubtract 1 a))); eauto.
intuition.
apply ratMult_0 in H3.
intuition.
eapply H0.
eapply ratSubtract_0_inv.
trivial.
edestruct (@expRat_le_exp_exists (a + (1 / 2) * (ratSubtract 1 a)) epsilon).
eapply half_distance_1_le; trivial.
trivial.
exists (Max.max x x0).
intuition.
unfold expRat_rel in *.
destruct (H1 n').
rewrite H6; eauto.
apply ratDistance_0_r_le.
eapply leRat_trans.
eapply expRat_leRat_compat.
eapply ratDistance_le_sum.
eapply H3.
eapply Max.max_lub_l.
eauto.
trivial.
eapply expRat_le'.
eauto.
eapply half_distance_1_le; trivial.
eapply Max.max_lub_r.
eauto.
Qed.
intuition.
eapply (@rat_inf_limit_trans _ (fun n => (ratMult_rel (ratSubtract_rel (eqRat 1) (expRat_rel (f n) n)) (ratInverse_rel (ratSubtract_rel (eqRat 1) (f n)))))).
unfold rat_inf_limit_2, inf_limit_2.
intuition.
exists (S O).
intuition.
destruct (H2 n').
eapply leRat_trans.
eapply eqRat_impl_leRat.
rewrite <- ratIdentityIndiscernables.
eapply sum_power_series.
assert (n' > O).
omega.
eapply H9.
econstructor.
eapply H8.
eapply H3.
intuition.
eapply H0.
eapply H9.
trivial.
trivial.
trivial.
eapply rat0_le_all.
eapply rat_inf_limit_eq.
eapply rat_inf_limit_product.
unfold left_total, ratSubtract_rel, expRat_rel. intuition.
destruct (H2 n).
exists (ratSubtract 1 (expRat x n)).
intuition.
eapply ratSubtract_eqRat_compat; eauto.
symmetry.
eapply H6.
trivial.
unfold left_total, ratInverse_rel, ratSubtract_rel. intuition.
destruct (H2 n).
exists (ratInverse (ratSubtract 1 x)).
intuition.
eapply ratInverse_eqRat_compat.
intuition.
eapply H0.
eauto.
eapply ratSubtract_0_inv.
trivial.
symmetry.
eapply H5; intuition.
eapply rat_inf_limit_difference.
unfold left_total; intuition.
exists 1; intuition.
unfold left_total, expRat_rel; intuition.
destruct (H2 n).
exists (expRat x n).
intuition.
eapply expRat_eqRat_compat; eauto.
eapply rat_inf_limit_const.
eapply eqRat_refl.
intuition.
rewrite <- H4.
rewrite H5.
intuition.
eapply rat_inf_limit_exp_0.
eauto.
trivial.
trivial.
eapply rat_inf_limit_ratInverse.
eapply rat_inf_limit_difference.
unfold left_total; intuition.
econstructor.
eapply eqRat_refl.
unfold left_total; intuition.
eapply rat_inf_limit_const.
eapply eqRat_refl.
intuition.
rewrite <- H4.
rewrite H5.
intuition.
eauto.
intuition.
eapply H1.
eapply ratSubtract_0_inv.
trivial.
eapply eqRat_refl.
unfold left_total; intuition.
destruct (H2 n).
exists (ratSubtract 1 x).
unfold ratSubtract_rel; intuition.
eapply ratSubtract_eqRat_compat; eauto.
rewrite ratSubtract_0_r.
rewrite ratMult_1_l.
intuition.
unfold left_total, ratMult_rel, ratSubtract_rel, expRat_rel, ratInverse_rel.
intuition.
destruct (H2 n).
exists ((ratSubtract 1 (expRat x n)) * (ratInverse (ratSubtract 1 x))).
intuition.
eapply ratMult_eqRat_compat.
symmetry.
eapply H5;
intuition.
eapply expRat_eqRat_compat.
eauto.
symmetry.
eapply H6.
intuition.
eapply ratSubtract_eqRat_compat; intuition.
eauto.
Qed.
unfold rat_inf_limit, inf_limit.
intuition.
destruct (H epsilon); intuition.
destruct (H1 x).
econstructor.
intuition.
eapply (H3 (g n')).
eapply le_trans.
2:{
eapply H0.
eapply H5.
}
rewrite <- H4.
eapply le_refl.
trivial.
Qed.

Lemma rat_inf_limit_sqrt:
forall (f : nat -> Rat -> Prop) (v : Rat),
rat_inf_limit f v ->
rat_inf_limit (fun n => f (Nat.sqrt n)) v.