CoRN.algebra.CFields

Require Export CRings.
Set Automatic Introduction.

Transparent sym_eq.
Transparent f_equal.

Transparent cs_crr.
Transparent csg_crr.
Transparent cm_crr.
Transparent cg_crr.
Transparent cr_crr.

Transparent csf_fun.
Transparent csbf_fun.
Transparent csr_rel.

Transparent cs_eq.
Transparent cs_neq.
Transparent cs_ap.
Transparent cm_unit.
Transparent csg_op.
Transparent cg_inv.
Transparent cg_minus.
Transparent cr_one.
Transparent cr_mult.

Transparent nexp_op.

Fields

Definition of the notion Field

Definition is_CField (R : CRing) (cf_rcpcl : ∀ x : R, x [#] [0] → R) : Prop :=
  ∀ x Hx, is_inverse cr_mult [1] x (cf_rcpcl x Hx).

Record CField : Type :=
  {cf_crr :> CRing;
   cf_rcpcl : ∀ x : cf_crr, x [#] [0] → cf_crr;
   cf_proof : is_CField cf_crr cf_rcpcl;
   cf_rcpsx : ∀ x y x_ y_, cf_rcpcl x x_ [#] cf_rcpcl y y_ → x [#] y}.

Definition f_rcpcl' (F : CField) : PartFunct F.
Proof.
apply Build_PartFunct with (fun x : F ⇒ x [#] [0]) (cf_rcpcl F).
  red in |- *; intros; astepl x. auto.
  exact (cf_rcpsx F).
Defined.

Definition f_rcpcl F x x_ := f_rcpcl' F x x_.

Implicit Arguments f_rcpcl [F].

cf_div is the division in a field. It is defined in terms of multiplication and the reciprocal. x/y is only defined if we have a proof of y [#] [0].

Definition cf_div (F : CField) (x y : F) y_ : F := x [*]f_rcpcl y y_.

Implicit Arguments cf_div [F].
Notation "x [/] y [//] Hy" := (cf_div x y Hy) (at level 80).

Division in fields is a (type dependent) ternary function: (cf_div x y Hy) is denoted infix by x / y ‡ Hy.
In lemmas, a hypothesis that t [#] [0] will be named t_.
We do not use Non[0]s, but write the condition [#] [0] separately.
In each lemma, we use only variables for proof objects, and these variables are universally quantified.

For example, the informal lemma (1/x).(1/y) = 1/(x.y) for all x and yis formalized as
∀ (x y : F) x_ y_ xy_, (1[/]x‡x_) [*] (1[/]y‡y_) [=] 1[/] (x[*]y)[//]xy_

and not as
∀ (x y : F) x_ y_, (1[/]x‡x_) [*] (1[/]y‡y_) [=] 1[/] (x[*]y)[//](prod_nz x y x_ y_)

We have made this choice to make it easier to apply lemmas; this can be quite awkward if we would use the last formulation.

So every division occurring in the formulation of a lemma is of the

form e/e'‡H where H is a variable. Only exceptions: we may write e/ (Snring n) and e/TwoNZ, e/ThreeNZ and so on. (Constants like TwoNZ will be defined later on.)

Field axioms

Let F be a field.

Section Field_axioms.
Variable F : CField.

Lemma CField_is_CField : is_CField F (cf_rcpcl F).
Proof.
elim F; auto.
Qed.

Lemma rcpcl_is_inverse : ∀ x x_, is_inverse cr_mult [1] x (cf_rcpcl F x x_).
Proof.
apply CField_is_CField.
Qed.

End Field_axioms.

Section Field_basics.

Field basics

Let F be a field.

Variable F : CField.

Lemma rcpcl_is_inverse_unfolded : ∀ x x_, x [*]cf_rcpcl F x x_ [=] [1].
Proof.
intros x x_.
elim (rcpcl_is_inverse F x x_); auto.
Qed.

Lemma field_mult_inv : ∀ (x : F) x_, x [*]f_rcpcl x x_ [=] [1].
Proof rcpcl_is_inverse_unfolded.
Hint Resolve field_mult_inv: algebra.

Lemma field_mult_inv_op : ∀ (x : F) x_, f_rcpcl x x_[*]x [=] [1].
Proof.
intros x x_.
elim (rcpcl_is_inverse F x x_); auto.
Qed.

End Field_basics.

Hint Resolve field_mult_inv field_mult_inv_op: algebra.

Section Field_multiplication.

Properties of multiplication

Let F be a field.

Variable F : CField.

Lemma mult_resp_ap_zero : ∀ x y : F, x [#] [0] → y [#] [0] → x [*]y [#] [0].
Proof.
intros x y Hx Hy.
apply cring_mult_ap_zero with (f_rcpcl y Hy).
astepl x.
  auto.
astepl (x[*][1]).
eapply eq_transitive_unfolded.
  2: apply CRings.mult_assoc.
algebra.
Qed.

Lemma mult_lft_resp_ap : ∀ x y z : F, x [#] y → z [#] [0] → z [*]x [#] z [*]y.
Proof.
intros x y z H Hz.
apply zero_minus_apart.
unfold cg_minus in |- ×.
astepl (z[*]x[+]z[*][--]y).
astepl (z[*] (x[+][--]y)).
astepl (z[*] (x[-]y)).
apply mult_resp_ap_zero; algebra.
Qed.

Lemma mult_rht_resp_ap : ∀ x y z : F, x [#] y → z [#] [0] → x [*]z [#] y [*]z.
Proof.
intros x y z H Hz.
astepl (z[*]x).
astepr (z[*]y).
apply mult_lft_resp_ap; assumption.
Qed.

Lemma mult_resp_neq_zero : ∀ x y : F, x [~=][0] → y [~=][0] → x [*]y [~=][0].
Proof.
intros x y Hx Hy.
cut (¬ Not (x [#] [0])).
  intro H.
  cut (¬ Not (y [#] [0])).
   intro H0.
   apply notnot_ap_imp_neq.
   cut (x [#] [0] → y [#] [0] → x[*]y [#] [0]).
    intro H1.
    intro.
    apply H0; intro H3.
    apply H; intro H4.
    apply H2; auto.
   intros; apply mult_resp_ap_zero; auto.
  apply neq_imp_notnot_ap; auto.
apply neq_imp_notnot_ap; auto.
Qed.

Lemma mult_resp_neq : ∀ x y z : F, x [~=]y → z [~=][0] → x [*]z [~=]y [*]z.
Proof.
intros x y z H Hz.
generalize (neq_imp_notnot_ap _ _ _ H).
generalize (neq_imp_notnot_ap _ _ _ Hz).
generalize (mult_rht_resp_ap x y z).
intros H1 H2 H3.
apply notnot_ap_imp_neq.
intro H4.
apply H2; intro.
apply H3; intro.
apply H4.
auto.
Qed.

Lemma mult_eq_zero : ∀ x y : F, x [~=][0] → x [*]y [=] [0] → y [=] [0].
Proof.
intros x y Hx Hxy.
apply not_ap_imp_eq.
intro H.
elim (eq_imp_not_neq _ _ _ Hxy).
apply mult_resp_neq_zero.
  assumption.
apply ap_imp_neq.
assumption.
Qed.

Lemma mult_cancel_lft : ∀ x y z : F, z [#] [0] → z [*]x [=] z [*]y → x [=] y.
Proof.
intros x y z Hz H.
apply not_ap_imp_eq.
intro H2.
elim (eq_imp_not_ap _ _ _ H).
apply mult_lft_resp_ap; assumption.
Qed.

Lemma mult_cancel_rht : ∀ x y z : F, z [#] [0] → x [*]z [=] y [*]z → x [=] y.
Proof.
intros x y z Hz H.
apply (mult_cancel_lft x y z).
  assumption.
astepr (y[*]z).
Step_final (x[*]z).
Qed.

Lemma square_eq_aux : ∀ x a : F, x [^]2 [=] a [^]2 → (x [+]a ) [*] (x [-]a ) [=] [0].
Proof.
intros x a H.
astepl (x[^]2[-]a[^]2).
Step_final (a[^]2[-]a[^]2).
Qed.

Lemma square_eq_weak : ∀ x a : F, x [^]2 [=] a [^]2 → Not (x [#] a and x [#] [--]a).
Proof.
intros x a H.
intro H0.
elim H0; intros H1 H2.
generalize (square_eq_aux _ _ H); intro H3.
generalize (eq_imp_not_ap _ _ _ H3); intro H4.
apply H4.
apply mult_resp_ap_zero.
  astepr ([--]a[+]a). apply op_rht_resp_ap. auto.
  astepr (a[-]a).
apply minus_resp_ap_rht.
assumption.
Qed.

Lemma cond_square_eq : ∀ x a : F,
(Two:F ) [#] [0] → a [#] [0] → x [^]2 [=] a [^]2 → x [=] a or x [=] [--]a.
Proof.
intros x a H Ha H0.
cut (a [#] [--]a).
  intro H1.
  elim (ap_cotransitive_unfolded _ _ _ H1 x); intro H2.
   right.
   apply not_ap_imp_eq.
   intro H3.
   elim (square_eq_weak _ _ H0).
   split; auto.
   apply ap_symmetric_unfolded; auto.
  left.
  apply not_ap_imp_eq.
  intro H3.
  elim (square_eq_weak _ _ H0); auto.
apply plus_cancel_ap_lft with a.
astepr ([0]:F).
astepl (Two [*]a).
apply mult_resp_ap_zero; auto.
Qed.
End Field_multiplication.

Section x_square.
Lemma x_xminone : ∀ (F : CField) (x : F), x [^]2 [=] x → x [*] (x [-][1]) [=] [0].
Proof.
intros H x h.
astepl (x[*]x[-]x[*][1]).
astepl (x[*]x[-]x).
apply cg_cancel_rht with x.
astepl (x[*]x[+][--]x[+]x).
astepl (x[*]x[+]([--]x[+]x)).
astepl (x[*]x[+][0]).
astepl (x[*]x).
astepr x.
astepl (x[^]2).
exact h.
Qed.

Lemma square_id : ∀ (F : CField) (x : F), x [^]2 [=] x → {x [=] [0]} + {x [=] [1]}.
Proof.
intros F x H.
cut (([0]:F) [#] ([1]:F)).
  intro H0.
  elim (ap_cotransitive_unfolded _ _ _ H0 x).
   intro H1.
   right.
   apply not_ap_imp_eq.
   red in |- ×.
   intro H2.
   set (H3 := minus_resp_ap_rht F x [1] [1] H2) in ×.
   set (H4 := ap_wdr_unfolded F (x[-][1]) ([1][-][1]) [0] H3 (cg_minus_correct F [1])) in ×.
   set (H5 := ap_symmetric_unfolded F [0] x H1) in ×.
   set (H6 := mult_resp_ap_zero F x (x[-][1]) H5 H4) in ×.
   simpl in |- ×.
   set (H7 := x_xminone F x H) in ×.
   set (H8 := eq_imp_not_ap F (x[*] (x[-][1])) [0] H7) in ×.
   intuition.
  left.
  apply not_ap_imp_eq.
  red in |- ×.
  intro H2.
  set (H3 := minus_resp_ap_rht F x [1] [1] b) in ×.
  set (H4 := ap_wdr_unfolded F (x[-][1]) ([1][-][1]) [0] H3 (cg_minus_correct F [1])) in ×.
  set (H6 := mult_resp_ap_zero F x (x[-][1]) H2 H4) in ×.
  set (H7 := x_xminone F x H) in ×.
  set (H8 := eq_imp_not_ap F (x[*] (x[-][1])) [0] H7) in ×.
  intuition.
apply ap_symmetric_unfolded.
apply ring_non_triv.
Qed.
End x_square.

Hint Resolve mult_resp_ap_zero: algebra.

Section Rcpcl_properties.

Properties of reciprocal

Let F be a field.

Variable F : CField.

Lemma inv_one : f_rcpcl [1] (ring_non_triv F) [=] [1].
Proof.
astepl ([1][*]f_rcpcl [1] (ring_non_triv F)).
apply field_mult_inv.
Qed.

Lemma f_rcpcl_wd : ∀ (x y : F) x_ y_, x [=] y → f_rcpcl x x_ [=] f_rcpcl y y_.
Proof.
intros x y H.
unfold f_rcpcl in |- *; algebra.
Qed.

Lemma f_rcpcl_mult : ∀ (y z : F) y_ z_ yz_,
f_rcpcl (y [*]z) yz_ [=] f_rcpcl y y_[*]f_rcpcl z z_.
Proof.
intros y z nzy nzz nzyz.
apply mult_cancel_lft with (y[*]z).
assumption.
astepl ([1]:F).
astepr (y[*]z[*] (f_rcpcl z nzz[*]f_rcpcl y nzy)).
astepr (y[*] (z[*] (f_rcpcl z nzz[*]f_rcpcl y nzy))).
astepr (y[*] (z[*]f_rcpcl z nzz[*]f_rcpcl y nzy)).
astepr (y[*] ([1][*]f_rcpcl y nzy)).
astepr (y[*]f_rcpcl y nzy).
Step_final ([1]:F).
Qed.

Lemma f_rcpcl_resp_ap_zero : ∀ (y : F) y_, f_rcpcl y y_ [#] [0].
Proof.
intros y nzy.
apply cring_mult_ap_zero_op with y.
astepl ([1]:F). apply one_ap_zero.
Qed.

Lemma f_rcpcl_f_rcpcl : ∀ (x : F) x_ r_, f_rcpcl (f_rcpcl x x_) r_ [=] x.
Proof.
intros x nzx nzr.
apply mult_cancel_rht with (f_rcpcl x nzx).
assumption.
astepr ([1]:F).
Step_final (f_rcpcl x nzx[*]f_rcpcl (f_rcpcl x nzx) nzr).
Qed.

End Rcpcl_properties.

Section MultipGroup.

The multiplicative group of nonzeros of a field.

Let F be a field

Variable F : CField.

The multiplicative monoid of Non0s.

Definition NonZeroMonoid : CMonoid := Build_SubCMonoid
(Build_multCMonoid F) (nonZeroP (M:=F)) (one_ap_zero F) (mult_resp_ap_zero F).

Definition fmg_cs_inv : CSetoid_un_op NonZeroMonoid.
Proof.
red in |- ×.
cut (∀ x : NonZeroMonoid, nonZeroP (cf_rcpcl F (scs_elem _ _ x) (scs_prf _ _ x))).
  intro H.
  apply Build_CSetoid_fun with (fun x : NonZeroMonoid ⇒
    Build_subcsetoid_crr _ _ (cf_rcpcl F (scs_elem _ _ x) (scs_prf _ _ x)) (H x)).
  red in |- ×.
  simpl in |- ×.
  simple destruct x; simple destruct y. intros scs_elem0 scs_prf0 H0.
  apply (cf_rcpsx _ _ _ _ _ H0).
intro; simpl in |- ×.
red in |- ×.
astepl (f_rcpcl (scs_elem _ _ x) (scs_prf _ _ x)).
apply f_rcpcl_resp_ap_zero.
Defined.

Lemma plus_nonzeros_eq_mult_dom : ∀ x y : NonZeroMonoid,
scs_elem _ _ (x [+]y) [=] scs_elem _ _ x [*]scs_elem _ _ y.
Proof.
simple destruct x; simple destruct y; algebra.
Qed.

Lemma cfield_to_mult_cgroup : CGroup.
Proof.
apply (Build_CGroup NonZeroMonoid fmg_cs_inv).
intro x.
red in |- ×.
elim x; intros x_ Hx.
simpl in |- *; apply cf_proof.
Qed.

End MultipGroup.

Section Div_properties.

Properties of division

Let F be a field.

In the names of lemmas, we denote / by div, and [1][/] by recip.

Variable F : CField.

Lemma div_prop : ∀ (x : F) x_, ([0]/ x ‡x_) [=] [0].
Proof.
unfold cf_div in |- *; algebra.
Qed.

Lemma div_1 : ∀ (x y : F) y_, (x / y ‡y_) [*]y [=] x.
Proof.
intros x y y_.
astepl (x[*]f_rcpcl y y_[*]y).
astepl (x[*] (f_rcpcl y y_[*]y)).
Step_final (x[*][1]).
Qed.

Lemma div_1' : ∀ (x y : F) y_, y [*] (x / y ‡y_) [=] x.
Proof.
intros x y y_.
astepl ((x/ y‡y_) [*]y).
apply div_1.
Qed.

Lemma div_1'' : ∀ (x y : F) y_, (x [*]y / y ‡y_) [=] x.
Proof.
intros x y y_.
unfold cf_div in |- ×.
astepl (y[*]x[*]f_rcpcl y y_).
astepl (y[*] (x[*]f_rcpcl y y_)).
change (y[*] (x/ y‡y_) [=] x) in |- ×.
apply div_1'.
Qed.

Hint Resolve div_1: algebra.

Lemma x_div_x : ∀ (x : F) x_, (x / x ‡x_) [=] [1].
Proof.
intros x x_.
unfold cf_div in |- ×.
apply field_mult_inv.
Qed.

Hint Resolve x_div_x: algebra.

Lemma x_div_one : ∀ x : F, (x / [1]‡ring_non_triv F ) [=] x.
Proof.
intro x.
unfold cf_div in |- ×.
generalize inv_one; intro H.
astepl (x[*][1]).
apply mult_one.
Qed.

The next lemma says x.(y/z) = (x.y)/z.

Lemma x_mult_y_div_z : ∀ (x y z : F) z_, x [*] (y / z ‡z_) [=] (x [*]y / z ‡z_).
Proof.
unfold cf_div in |- *; algebra.
Qed.

Hint Resolve x_mult_y_div_z: algebra.

Lemma div_wd : ∀ (x x' y y' : F) y_ y'_, x [=] x' → y [=] y' → (x / y ‡y_) [=] (x'/ y'‡y'_).
Proof.
intros x x' y y' nzy nzy' H H0.
unfold cf_div in |- ×.
cut (f_rcpcl y nzy [=] f_rcpcl y' nzy').
intro H1.
algebra.
apply f_rcpcl_wd.
assumption.
Qed.

Hint Resolve div_wd: algebra_c.

The next lemma says [(x/y)/z = x/(y.z)]

Lemma div_div : ∀ (x y z : F) y_ z_ yz_, ((x / y ‡y_) / z ‡z_) [=] (x / y [*]z ‡yz_).
Proof.
intros x y z nzy nzz nzyz.
unfold cf_div in |- ×.
astepl (x[*] (f_rcpcl y nzy[*]f_rcpcl z nzz)).
apply mult_wdr.
apply eq_symmetric_unfolded.
apply f_rcpcl_mult.
Qed.

Lemma div_resp_ap_zero_rev : ∀ (x y : F) y_, x [#] [0] → (x / y ‡y_) [#] [0].
Proof.
intros x y nzy Hx.
unfold cf_div in |- ×.
apply mult_resp_ap_zero.
assumption.
apply f_rcpcl_resp_ap_zero.
Qed.

Lemma div_resp_ap_zero : ∀ (x y : F) y_, (x / y ‡y_) [#] [0] → x [#] [0].
Proof.
intros x y nzy Hxy.
astepl ((x/ y‡nzy) [*]y). algebra.
Qed.

The next lemma says [x/(y/z) = (x.z)/y]

Lemma div_div2 : ∀ (x y z : F) y_ z_ yz_, (x / y / z ‡z_‡yz_) [=] (x [*]z / y ‡y_).
Proof.
intros x y z nzy nzz nzyz.
unfold cf_div in |- ×.
astepr (x[*] (z[*]f_rcpcl y nzy)).
apply mult_wdr.
cut (f_rcpcl z nzz [#] [0]).
  intro nzrz.
  apply eq_transitive_unfolded with (f_rcpcl y nzy[*]f_rcpcl (f_rcpcl z nzz) nzrz).
   apply f_rcpcl_mult.
  astepr (f_rcpcl y nzy[*]z).
  apply mult_wdr.
  apply f_rcpcl_f_rcpcl.
apply f_rcpcl_resp_ap_zero.
Qed.

The next lemma says [(x.p)/(y.q) = (x/y).(p/q)]

Lemma mult_of_divs : ∀ (x y p q : F) y_ q_ yq_,
(x [*]p / y [*]q ‡yq_) [=] (x / y ‡y_) [*] (p / q ‡q_).
Proof.
intros x y p q nzy nzq nzyq.
unfold cf_div in |- ×.
astepl (x[*] (p[*]f_rcpcl (y[*]q) nzyq)).
astepr (x[*] (f_rcpcl y nzy[*] (p[*]f_rcpcl q nzq))).
apply mult_wdr.
astepr (f_rcpcl y nzy[*]p[*]f_rcpcl q nzq).
astepr (p[*]f_rcpcl y nzy[*]f_rcpcl q nzq).
astepr (p[*] (f_rcpcl y nzy[*]f_rcpcl q nzq)).
apply mult_wdr.
apply f_rcpcl_mult.
Qed.

Lemma div_dist : ∀ (x y z : F) z_, (x [+]y / z ‡z_) [=] (x / z ‡z_) [+] (y / z ‡z_).
Proof.
unfold cf_div in |- *; algebra.
Qed.

Lemma div_dist' : ∀ (x y z : F) z_, (x [-]y / z ‡z_) [=] (x / z ‡z_) [-] (y / z ‡z_).
Proof.
unfold cf_div in |- *; algebra.
Qed.

Lemma div_semi_sym : ∀ (x y z : F) y_ z_, ((x / y ‡y_) / z ‡z_) [=] ((x / z ‡z_) / y ‡y_).
Proof.
intros.
unfold cf_div in |- ×.
astepl (x[*] ((f_rcpcl y y_) [*] (f_rcpcl z z_))).
Step_final (x[*] ((f_rcpcl z z_) [*] (f_rcpcl y y_))).
Qed.

Hint Resolve div_semi_sym: algebra.

Lemma eq_div : ∀ (x y u v : F) y_ v_, x [*]v [=] u [*]y → (x / y ‡y_) [=] (u / v ‡v_).
Proof.
intros x y u v Hy Hv H.
astepl (x[*][1]/ y‡Hy).
astepl (x[*] (v/ v‡Hv) / y‡Hy).
astepl ((x[*]v/ v‡Hv) / y‡Hy).
astepl ((u[*]y/ v‡Hv) / y‡Hy).
astepl ((u[*]y/ y‡Hy) / v‡Hv).
astepl (u[*] (y/ y‡Hy) / v‡Hv).
Step_final (u[*][1]/ v‡Hv).
Qed.

Lemma div_strext : ∀ (x x' y y' : F) y_ y'_, (x / y ‡y_) [#] (x'/ y'‡y'_) → x [#] x' or y [#] y'.
Proof.
intros x x' y y' Hy Hy' H.
unfold cf_div in H.
elim (cs_bin_op_strext F cr_mult _ _ _ _ H).
  auto.
intro H1.
right.
unfold f_rcpcl in H1.
exact (pfstrx _ _ _ _ _ _ H1).
Qed.

End Div_properties.

Hint Resolve div_1 div_1' div_1'' div_wd x_div_x x_div_one div_div div_div2
  mult_of_divs x_mult_y_div_z mult_of_divs div_dist div_dist' div_semi_sym
  div_prop: algebra.

Cancellation laws for apartness and multiplication

Let F be a field

Section Mult_Cancel_Ap_Zero.

Variable F : CField.

Lemma mult_cancel_ap_zero_lft : ∀ x y : F, x [*]y [#] [0] → x [#] [0].
Proof.
intros x y H.
cut (x[*]y [#] [0][*][0]).
  intro H0.
  elim (bin_op_strext_unfolded _ _ _ _ _ _ H0); intro H1.
   3: astepr ([0]:F); auto.
  assumption.
astepl (x[*]y/ y‡H1).
apply div_resp_ap_zero_rev.
assumption.
Qed.

Lemma mult_cancel_ap_zero_rht : ∀ x y : F, x [*]y [#] [0] → y [#] [0].
Proof.
intros x y H.
apply mult_cancel_ap_zero_lft with x.
astepl (x[*]y). auto.
Qed.

Lemma recip_ap_zero : ∀ (x : F) x_, ([1]/ x ‡x_) [#] [0].
Proof.
intros; apply cring_mult_ap_zero with x.
astepl ([1]:F). algebra.
Qed.

Lemma recip_resp_ap : ∀ (x y : F) x_ y_, x [#] y → ([1]/ x ‡x_) [#] ([1]/ y ‡y_).
Proof.
intros x y x_ y_ H.
apply zero_minus_apart.
apply mult_cancel_ap_zero_lft with (x[*]y).
apply ap_wdl with (y[-]x).
  apply minus_ap_zero.
  apply ap_symmetric_unfolded; assumption.
eapply eq_transitive_unfolded.
  2: apply eq_symmetric_unfolded; apply dist_2b.
apply cg_minus_wd.
  astepr (x[*]y[*] ([1]/ x‡x_)).
  astepr (x[*]y[*][1]/ x‡x_).
  astepr (x[*]y/ x‡x_).
  astepr (y[*]x/ x‡x_).
  astepr (y[*] (x/ x‡x_)).
  Step_final (y[*][1]).
astepr (x[*]y[*] ([1]/ y‡y_)).
astepr (x[*]y[*][1]/ y‡y_).
astepr (x[*]y/ y‡y_).
astepr (x[*] (y/ y‡y_)).
Step_final (x[*][1]).
Qed.

End Mult_Cancel_Ap_Zero.

Section CField_Ops.

Functional Operations

We now move on to lifting these operations to functions. As we are dealing with partial functions, we don't have to worry explicitly about the function by which we are dividing being non-zero everywhere; this will simply be encoded in its domain.

Let X be a Field and F,G:(PartFunct X) have domains respectively P and Q.

Variable X : CField.

Variables F G : PartFunct X.

Section Part_Function_Recip.

Some auxiliary notions are helpful in defining the domain.

Let R := extend Q (fun x Hx ⇒ G x Hx [#] [0]).

Let Ext2R := ext2 (S:=X) (P:=Q) (R:=fun x Hx ⇒ G x Hx [#] [0]).

Lemma part_function_recip_strext : ∀ x y Hx Hy,
([1]/ _‡Ext2R x Hx ) [#] ([1]/ _‡Ext2R y Hy ) → x [#] y.
Proof.
intros x y Hx Hy H.
elim (div_strext _ _ _ _ _ _ _ H); intro H1.
elimtype False; apply ap_irreflexive_unfolded with (x := [1]:X); auto.
exact (pfstrx _ _ _ _ _ _ H1).
Qed.

Lemma part_function_recip_pred_wd : pred_wd X R.
Proof.
red in |- *; intros x y H H0.
elim H; intros H1 H2; split.
apply (dom_wd X G x y H1 H0).
intro H3; astepl (G x H1). auto.
Qed.

Definition Frecip := Build_PartFunct X _ part_function_recip_pred_wd
(fun x Hx ⇒ [1]/ _‡Ext2R x Hx) part_function_recip_strext.

End Part_Function_Recip.

Section Part_Function_Div.

For division things work out almost in the same way.

Let R := Conj P (extend Q (fun x Hx ⇒ G x Hx [#] [0])).

Let Ext2R := ext2 (S:=X) (P:=Q) (R:=fun x Hx ⇒ G x Hx [#] [0]).

Lemma part_function_div_strext : ∀ x y Hx Hy,
(F x (prj1 X _ _ _ Hx) / _‡Ext2R x (prj2 X _ _ _ Hx)) [#]
  (F y (prj1 X _ _ _ Hy) / _‡Ext2R y (prj2 X _ _ _ Hy)) →
x [#] y.
Proof.
intros x y Hx Hy H.
elim (div_strext _ _ _ _ _ _ _ H); intro H1; exact (pfstrx _ _ _ _ _ _ H1).
Qed.

Lemma part_function_div_pred_wd : pred_wd X R.
Proof.
red in |- *; intros x y H H0.
elim H; intros H1 H2; split.
  apply (dom_wd X F x y H1 H0).
clear H1.
elim H2; intros H1 H3; split.
  apply (dom_wd X G x y H1 H0).
intro H4; astepl (G x H1). auto.
Qed.

Definition Fdiv := Build_PartFunct X _ part_function_div_pred_wd
(fun x Hx ⇒ F x (Prj1 Hx) / _‡Ext2R x (Prj2 Hx)) part_function_div_strext.

End Part_Function_Div.

Let R:X→CProp.

Variable R:X → CProp.

Lemma included_FRecip : included R Q →
(∀ x, R x → ∀ Hx, G x Hx [#] [0]) → included R (Dom Frecip).
Proof.
intros H H0.
simpl in |- ×.
unfold extend in |- ×.
split.
  apply H; assumption.
intros; apply H0; assumption.
Qed.

Lemma included_FRecip' : included R (Dom Frecip) → included R Q.
Proof.
intro H; simpl in H; eapply included_extend; apply H.
Qed.

Lemma included_FDiv : included R P → included R Q →
(∀ x, R x → ∀ Hx, G x Hx [#] [0]) → included R (Dom Fdiv).
Proof.
intros HP HQ Hx.
simpl in |- ×.
apply included_conj.
  assumption.
unfold extend in |- ×.
split.
  apply HQ; assumption.
intros; apply Hx; assumption.
Qed.

Lemma included_FDiv' : included R (Dom Fdiv) → included R P.
Proof.
intro H; simpl in H; eapply included_conj_lft; apply H.
Qed.

Lemma included_FDiv'' : included R (Dom Fdiv) → included R Q.
intro H; simpl in H; eapply included_extend; eapply included_conj_rht; apply H.
Qed.

End CField_Ops.

Implicit Arguments Frecip [X].
Notation "{1/} x" := (Frecip x) (at level 4, right associativity).

Implicit Arguments Fdiv [X].
Infix "/" := Fdiv (at level 41, no associativity).

Hint Resolve included_FRecip included_FDiv : included.

Hint Immediate included_FRecip' included_FDiv' included_FDiv'' : included.