--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/fields/".
+
+include "rings.ma".
+
+record is_field (R:ring) (inv:∀x:R.x ≠ 0 → R) : Prop
+≝
+ { (* multiplicative abelian properties *)
+ mult_comm_: symmetric ? (mult R);
+ (* multiplicative group properties *)
+ inv_inverse_: ∀x.∀p: x ≠ 0. mult ? (inv x p) x = 1
+ }.
+
+lemma opp_opp: \forall R:ring. \forall x:R. (-(-x))=x.
+intros;
+apply (cancellationlaw ? (-x) ? ?);
+rewrite > (opp_inverse R x);
+rewrite > plus_comm;
+rewrite > opp_inverse;
+reflexivity.
+qed.
+
+let rec sum (C:Type) (plus:C→C→C) (zero,one:C) (n:nat) on n ≝
+ match n with
+ [ O ⇒ zero
+ | (S m) ⇒ plus one (sum C plus zero one m)
+ ].
+
+record field : Type \def
+ { f_ring:> ring;
+ inv: ∀x:f_ring. x ≠ 0 → f_ring;
+ field_properties: is_field f_ring inv
+ }.
+
+theorem mult_comm: ∀F:field.symmetric ? (mult F).
+ intro;
+ apply (mult_comm_ ? ? (field_properties F)).
+qed.
+
+theorem inv_inverse: ∀F:field.∀x.∀p: x ≠ 0. mult ? (inv F x p) x = 1.
+ intro;
+ apply (inv_inverse_ ? ? (field_properties F)).
+qed.
+
+definition sum_field ≝
+ λF:field. sum ? (plus F) (zero F) (one F).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/groups/".
+
+include "higher_order_defs/functions.ma".
+include "nat/nat.ma".
+include "nat/orders.ma".
+
+definition left_neutral \def λC,op.λe:C. ∀x:C. op e x = x.
+
+definition right_neutral \def λC,op. λe:C. ∀x:C. op x e=x.
+
+definition left_inverse \def λC,op.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e.
+
+definition right_inverse \def λC,op.λe:C.λ inv: C→ C. ∀x:C. op x (inv x)=e.
+
+definition distributive_left ≝
+ λA:Type.λf:A→A→A.λg:A→A→A.
+ ∀x,y,z. f x (g y z) = g (f x y) (f x z).
+
+definition distributive_right ≝
+ λA:Type.λf:A→A→A.λg:A→A→A.
+ ∀x,y,z. f (g x y) z = g (f x z) (f y z).
+
+record is_abelian_group (C:Type) (plus:C→C→C) (zero:C) (opp:C→C) : Prop \def
+ { (* abelian additive semigroup properties *)
+ plus_assoc_: associative ? plus;
+ plus_comm_: symmetric ? plus;
+ (* additive monoid properties *)
+ zero_neutral_: left_neutral ? plus zero;
+ (* additive group properties *)
+ opp_inverse_: left_inverse ? plus zero opp
+ }.
+
+record abelian_group : Type \def
+ { carrier:> Type;
+ plus: carrier → carrier → carrier;
+ zero: carrier;
+ opp: carrier → carrier;
+ ag_abelian_group_properties: is_abelian_group ? plus zero opp
+ }.
+
+notation "0" with precedence 89
+for @{ 'zero }.
+
+interpretation "Abelian group zero" 'zero =
+ (cic:/matita/groups/zero.con _).
+
+interpretation "Abelian group plus" 'plus a b =
+ (cic:/matita/groups/plus.con _ a b).
+
+interpretation "Abelian group opp" 'uminus a =
+ (cic:/matita/groups/opp.con _ a).
+
+definition minus ≝
+ λG:abelian_group.λa,b:G. a + -b.
+
+interpretation "Abelian group minus" 'minus a b =
+ (cic:/matita/groups/minus.con _ a b).
+
+theorem plus_assoc: ∀G:abelian_group. associative ? (plus G).
+ intro;
+ apply (plus_assoc_ ? ? ? ? (ag_abelian_group_properties G)).
+qed.
+
+theorem plus_comm: ∀G:abelian_group. symmetric ? (plus G).
+ intro;
+ apply (plus_comm_ ? ? ? ? (ag_abelian_group_properties G)).
+qed.
+
+theorem zero_neutral: ∀G:abelian_group. left_neutral ? (plus G) 0.
+ intro;
+ apply (zero_neutral_ ? ? ? ? (ag_abelian_group_properties G)).
+qed.
+
+theorem opp_inverse: ∀G:abelian_group. left_inverse ? (plus G) 0 (opp G).
+ intro;
+ apply (opp_inverse_ ? ? ? ? (ag_abelian_group_properties G)).
+qed.
+
+lemma cancellationlaw: ∀G:abelian_group.∀x,y,z:G. x+y=x+z → y=z.
+intros;
+generalize in match (eq_f ? ? (λa.-x +a) ? ? H);
+intros; clear H;
+rewrite < plus_assoc in H1;
+rewrite < plus_assoc in H1;
+rewrite > opp_inverse in H1;
+rewrite > zero_neutral in H1;
+rewrite > zero_neutral in H1;
+assumption.
+qed.
+
set "baseuri" "cic:/matita/integration_algebras/".
-include "higher_order_defs/functions.ma".
-include "nat/nat.ma".
-include "nat/orders.ma".
-
-definition left_neutral \def λC,op.λe:C. ∀x:C. op e x = x.
-
-definition right_neutral \def λC,op. λe:C. ∀x:C. op x e=x.
-
-definition left_inverse \def λC,op.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e.
-
-definition right_inverse \def λC,op.λe:C.λ inv: C→ C. ∀x:C. op x (inv x)=e.
-
-definition distributive_left ≝
- λA:Type.λf:A→A→A.λg:A→A→A.
- ∀x,y,z. f x (g y z) = g (f x y) (f x z).
-
-definition distributive_right ≝
- λA:Type.λf:A→A→A.λg:A→A→A.
- ∀x,y,z. f (g x y) z = g (f x z) (f y z).
-
-record is_abelian_group (C:Type) (plus:C→C→C) (zero:C) (opp:C→C) : Prop \def
- { (* abelian additive semigroup properties *)
- plus_assoc_: associative ? plus;
- plus_comm_: symmetric ? plus;
- (* additive monoid properties *)
- zero_neutral_: left_neutral ? plus zero;
- (* additive group properties *)
- opp_inverse_: left_inverse ? plus zero opp
- }.
-
-record abelian_group : Type \def
- { carrier:> Type;
- plus: carrier → carrier → carrier;
- zero: carrier;
- opp: carrier → carrier;
- ag_abelian_group_properties: is_abelian_group ? plus zero opp
- }.
-
-notation "0" with precedence 89
-for @{ 'zero }.
-
-interpretation "Ring zero" 'zero =
- (cic:/matita/integration_algebras/zero.con _).
-
-interpretation "Ring plus" 'plus a b =
- (cic:/matita/integration_algebras/plus.con _ a b).
-
-interpretation "Ring opp" 'uminus a =
- (cic:/matita/integration_algebras/opp.con _ a).
-
-definition minus ≝
- λG:abelian_group.λa,b:G. a + -b.
-
-interpretation "Ring minus" 'minus a b =
- (cic:/matita/integration_algebras/minus.con _ a b).
-
-theorem plus_assoc: ∀G:abelian_group. associative ? (plus G).
- intro;
- apply (plus_assoc_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
-
-theorem plus_comm: ∀G:abelian_group. symmetric ? (plus G).
- intro;
- apply (plus_comm_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
-
-theorem zero_neutral: ∀G:abelian_group. left_neutral ? (plus G) 0.
- intro;
- apply (zero_neutral_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
-
-theorem opp_inverse: ∀G:abelian_group. left_inverse ? (plus G) 0 (opp G).
- intro;
- apply (opp_inverse_ ? ? ? ? (ag_abelian_group_properties G)).
-qed.
-
-lemma cancellationlaw: ∀G:abelian_group.∀x,y,z:G. x+y=x+z → y=z.
-intros;
-generalize in match (eq_f ? ? (λa.-x +a) ? ? H);
-intros; clear H;
-rewrite < plus_assoc in H1;
-rewrite < plus_assoc in H1;
-rewrite > opp_inverse in H1;
-rewrite > zero_neutral in H1;
-rewrite > zero_neutral in H1;
-assumption.
-qed.
-
-(****************************** rings *********************************)
-
-record is_ring (G:abelian_group) (mult:G→G→G) (one:G) : Prop
-≝
- { (* multiplicative monoid properties *)
- mult_assoc_: associative ? mult;
- one_neutral_left_: left_neutral ? mult one;
- one_neutral_right_: right_neutral ? mult one;
- (* ring properties *)
- mult_plus_distr_left_: distributive_left ? mult (plus G);
- mult_plus_distr_right_: distributive_right ? mult (plus G);
- not_eq_zero_one_: (0 ≠ one)
- }.
-
-record ring : Type \def
- { r_abelian_group:> abelian_group;
- mult: r_abelian_group → r_abelian_group → r_abelian_group;
- one: r_abelian_group;
- r_ring_properties: is_ring r_abelian_group mult one
- }.
-
-theorem mult_assoc: ∀R:ring.associative ? (mult R).
- intros;
- apply (mult_assoc_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem one_neutral_left: ∀R:ring.left_neutral ? (mult R) (one R).
- intros;
- apply (one_neutral_left_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem one_neutral_right: ∀R:ring.right_neutral ? (mult R) (one R).
- intros;
- apply (one_neutral_right_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem mult_plus_distr_left: ∀R:ring.distributive_left ? (mult R) (plus R).
- intros;
- apply (mult_plus_distr_left_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem mult_plus_distr_right: ∀R:ring.distributive_right ? (mult R) (plus R).
- intros;
- apply (mult_plus_distr_right_ ? ? ? (r_ring_properties R)).
-qed.
-
-theorem not_eq_zero_one: ∀R:ring.0 ≠ one R.
- intros;
- apply (not_eq_zero_one_ ? ? ? (r_ring_properties R)).
-qed.
-
-interpretation "Ring mult" 'times a b =
- (cic:/matita/integration_algebras/mult.con _ a b).
-
-notation "1" with precedence 89
-for @{ 'one }.
-
-interpretation "Field one" 'one =
- (cic:/matita/integration_algebras/one.con _).
-
-lemma eq_mult_zero_x_zero: ∀R:ring.∀x:R.0*x=0.
- intros;
- generalize in match (zero_neutral R 0); intro;
- generalize in match (eq_f ? ? (λy.y*x) ? ? H); intro; clear H;
- rewrite > mult_plus_distr_right in H1;
- generalize in match (eq_f ? ? (λy.-(0*x)+y) ? ? H1); intro; clear H1;
- rewrite < plus_assoc in H;
- rewrite > opp_inverse in H;
- rewrite > zero_neutral in H;
- assumption.
-qed.
-
-lemma eq_mult_x_zero_zero: ∀R:ring.∀x:R.x*0=0.
-intros;
-generalize in match (zero_neutral R 0);
-intro;
-generalize in match (eq_f ? ? (\lambda y.x*y) ? ? H); intro; clear H;
-rewrite > mult_plus_distr_left in H1;
-generalize in match (eq_f ? ? (\lambda y. (-(x*0)) +y) ? ? H1);intro;
-clear H1;
-rewrite < plus_assoc in H;
-rewrite > opp_inverse in H;
-rewrite > zero_neutral in H;
-assumption.
-qed.
-
-record is_field (R:ring) (inv:∀x:R.x ≠ 0 → R) : Prop
-≝
- { (* multiplicative abelian properties *)
- mult_comm_: symmetric ? (mult R);
- (* multiplicative group properties *)
- inv_inverse_: ∀x.∀p: x ≠ 0. mult ? (inv x p) x = 1
- }.
-
-lemma opp_opp: \forall R:ring. \forall x:R. (-(-x))=x.
-intros;
-apply (cancellationlaw ? (-x) ? ?);
-rewrite > (opp_inverse R x);
-rewrite > plus_comm;
-rewrite > opp_inverse;
-reflexivity.
-qed.
-
-
-let rec sum (C:Type) (plus:C→C→C) (zero,one:C) (n:nat) on n ≝
- match n with
- [ O ⇒ zero
- | (S m) ⇒ plus one (sum C plus zero one m)
- ].
-
-record field : Type \def
- { f_ring:> ring;
- inv: ∀x:f_ring. x ≠ 0 → f_ring;
- field_properties: is_field f_ring inv
- }.
-
-theorem mult_comm: ∀F:field.symmetric ? (mult F).
- intro;
- apply (mult_comm_ ? ? (field_properties F)).
-qed.
-
-theorem inv_inverse: ∀F:field.∀x.∀p: x ≠ 0. mult ? (inv F x p) x = 1.
- intro;
- apply (inv_inverse_ ? ? (field_properties F)).
-qed.
-
-definition sum_field ≝
- λF:field. sum ? (plus F) (zero F) (one F).
-
-record is_ordered_field_ch0 (F:field) (le:F→F→Prop) : Prop \def
- { of_mult_compat: ∀a,b. le 0 a → le 0 b → le 0 (a*b);
- of_plus_compat: ∀a,b,c. le a b → le (a+c) (b+c);
- of_weak_tricotomy : ∀a,b. a≠b → le a b ∨ le b a;
- (* 0 characteristics *)
- of_char0: ∀n. n > O → sum ? (plus F) 0 1 n ≠ 0
- }.
-
-record ordered_field_ch0 : Type \def
- { of_field:> field;
- of_le: of_field → of_field → Prop;
- of_ordered_field_properties:> is_ordered_field_ch0 of_field of_le
- }.
-
-interpretation "Ordered field le" 'leq a b =
- (cic:/matita/integration_algebras/of_le.con _ a b).
-
-definition lt \def λF:ordered_field_ch0.λa,b:F.a ≤ b ∧ a ≠ b.
-
-interpretation "Ordered field lt" 'lt a b =
- (cic:/matita/integration_algebras/lt.con _ a b).
-
-lemma le_zero_x_to_le_opp_x_zero: ∀F:ordered_field_ch0.∀x:F. 0 ≤ x → -x ≤ 0.
-intros;
- generalize in match (of_plus_compat ? ? F ? ? (-x) H); intro;
- rewrite > zero_neutral in H1;
- rewrite > plus_comm in H1;
- rewrite > opp_inverse in H1;
- assumption.
-qed.
-
-lemma le_x_zero_to_le_zero_opp_x: ∀F:ordered_field_ch0.∀x:F. x ≤ 0 → 0 ≤ -x.
- intros;
- generalize in match (of_plus_compat ? ? F ? ? (-x) H); intro;
- rewrite > zero_neutral in H1;
- rewrite > plus_comm in H1;
- rewrite > opp_inverse in H1;
- assumption.
-qed.
-
-(*
-lemma eq_opp_x_times_opp_one_x: ∀F:ordered_field_ch0.∀x:F.-x = -1*x.
- intros;
-
-lemma not_eq_x_zero_to_lt_zero_mult_x_x:
- ∀F:ordered_field_ch0.∀x:F. x ≠ 0 → 0 < x * x.
- intros;
- elim (of_weak_tricotomy ? ? ? ? ? ? ? ? F ? ? H);
- [ generalize in match (le_x_zero_to_le_zero_opp_x F ? H1); intro;
- generalize in match (of_mult_compat ? ? ? ? ? ? ? ? F ? ? H2 H2); intro;
-*)
-
-(* The ordering is not necessary. *)
-axiom not_eq_sum_field_zero: ∀F:ordered_field_ch0.∀n. O<n → sum_field F n ≠ 0.
+include "reals.ma".
record is_vector_space (K: field) (G:abelian_group) (emult:K→G→G) : Prop
≝
) * (a_one ? ? A)))) 0;
ifa_quotient_space1:
∀f,g:A. f=g → I(absolute_value ? A (f - g)) = 0
- }.
\ No newline at end of file
+ }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/ordered_fields_ch0/".
+
+include "fields.ma".
+
+record is_ordered_field_ch0 (F:field) (le:F→F→Prop) : Prop \def
+ { of_mult_compat: ∀a,b. le 0 a → le 0 b → le 0 (a*b);
+ of_plus_compat: ∀a,b,c. le a b → le (a+c) (b+c);
+ of_weak_tricotomy : ∀a,b. a≠b → le a b ∨ le b a;
+ (* 0 characteristics *)
+ of_char0: ∀n. n > O → sum ? (plus F) 0 1 n ≠ 0
+ }.
+
+record ordered_field_ch0 : Type \def
+ { of_field:> field;
+ of_le: of_field → of_field → Prop;
+ of_ordered_field_properties:> is_ordered_field_ch0 of_field of_le
+ }.
+
+interpretation "Ordered field le" 'leq a b =
+ (cic:/matita/ordered_fields_ch0/of_le.con _ a b).
+
+definition lt \def λF:ordered_field_ch0.λa,b:F.a ≤ b ∧ a ≠ b.
+
+interpretation "Ordered field lt" 'lt a b =
+ (cic:/matita/ordered_fields_ch0/lt.con _ a b).
+
+lemma le_zero_x_to_le_opp_x_zero: ∀F:ordered_field_ch0.∀x:F. 0 ≤ x → -x ≤ 0.
+intros;
+ generalize in match (of_plus_compat ? ? F ? ? (-x) H); intro;
+ rewrite > zero_neutral in H1;
+ rewrite > plus_comm in H1;
+ rewrite > opp_inverse in H1;
+ assumption.
+qed.
+
+lemma le_x_zero_to_le_zero_opp_x: ∀F:ordered_field_ch0.∀x:F. x ≤ 0 → 0 ≤ -x.
+ intros;
+ generalize in match (of_plus_compat ? ? F ? ? (-x) H); intro;
+ rewrite > zero_neutral in H1;
+ rewrite > plus_comm in H1;
+ rewrite > opp_inverse in H1;
+ assumption.
+qed.
+
+(*
+lemma eq_opp_x_times_opp_one_x: ∀F:ordered_field_ch0.∀x:F.-x = -1*x.
+ intros;
+
+lemma not_eq_x_zero_to_lt_zero_mult_x_x:
+ ∀F:ordered_field_ch0.∀x:F. x ≠ 0 → 0 < x * x.
+ intros;
+ elim (of_weak_tricotomy ? ? ? ? ? ? ? ? F ? ? H);
+ [ generalize in match (le_x_zero_to_le_zero_opp_x F ? H1); intro;
+ generalize in match (of_mult_compat ? ? ? ? ? ? ? ? F ? ? H2 H2); intro;
+*)
+
+(* The ordering is not necessary. *)
+axiom not_eq_sum_field_zero: ∀F:ordered_field_ch0.∀n. O<n → sum_field F n ≠ 0.
+
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/reals/".
+
+include "ordered_fields_ch0.ma".
+
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/rings/".
+
+include "groups.ma".
+
+record is_ring (G:abelian_group) (mult:G→G→G) (one:G) : Prop
+≝
+ { (* multiplicative monoid properties *)
+ mult_assoc_: associative ? mult;
+ one_neutral_left_: left_neutral ? mult one;
+ one_neutral_right_: right_neutral ? mult one;
+ (* ring properties *)
+ mult_plus_distr_left_: distributive_left ? mult (plus G);
+ mult_plus_distr_right_: distributive_right ? mult (plus G);
+ not_eq_zero_one_: (0 ≠ one)
+ }.
+
+record ring : Type \def
+ { r_abelian_group:> abelian_group;
+ mult: r_abelian_group → r_abelian_group → r_abelian_group;
+ one: r_abelian_group;
+ r_ring_properties: is_ring r_abelian_group mult one
+ }.
+
+theorem mult_assoc: ∀R:ring.associative ? (mult R).
+ intros;
+ apply (mult_assoc_ ? ? ? (r_ring_properties R)).
+qed.
+
+theorem one_neutral_left: ∀R:ring.left_neutral ? (mult R) (one R).
+ intros;
+ apply (one_neutral_left_ ? ? ? (r_ring_properties R)).
+qed.
+
+theorem one_neutral_right: ∀R:ring.right_neutral ? (mult R) (one R).
+ intros;
+ apply (one_neutral_right_ ? ? ? (r_ring_properties R)).
+qed.
+
+theorem mult_plus_distr_left: ∀R:ring.distributive_left ? (mult R) (plus R).
+ intros;
+ apply (mult_plus_distr_left_ ? ? ? (r_ring_properties R)).
+qed.
+
+theorem mult_plus_distr_right: ∀R:ring.distributive_right ? (mult R) (plus R).
+ intros;
+ apply (mult_plus_distr_right_ ? ? ? (r_ring_properties R)).
+qed.
+
+theorem not_eq_zero_one: ∀R:ring.0 ≠ one R.
+ intros;
+ apply (not_eq_zero_one_ ? ? ? (r_ring_properties R)).
+qed.
+
+interpretation "Ring mult" 'times a b =
+ (cic:/matita/rings/mult.con _ a b).
+
+notation "1" with precedence 89
+for @{ 'one }.
+
+interpretation "Ring one" 'one =
+ (cic:/matita/rings/one.con _).
+
+lemma eq_mult_zero_x_zero: ∀R:ring.∀x:R.0*x=0.
+ intros;
+ generalize in match (zero_neutral R 0); intro;
+ generalize in match (eq_f ? ? (λy.y*x) ? ? H); intro; clear H;
+ rewrite > mult_plus_distr_right in H1;
+ generalize in match (eq_f ? ? (λy.-(0*x)+y) ? ? H1); intro; clear H1;
+ rewrite < plus_assoc in H;
+ rewrite > opp_inverse in H;
+ rewrite > zero_neutral in H;
+ assumption.
+qed.
+
+lemma eq_mult_x_zero_zero: ∀R:ring.∀x:R.x*0=0.
+intros;
+generalize in match (zero_neutral R 0);
+intro;
+generalize in match (eq_f ? ? (\lambda y.x*y) ? ? H); intro; clear H;
+rewrite > mult_plus_distr_left in H1;
+generalize in match (eq_f ? ? (\lambda y. (-(x*0)) +y) ? ? H1);intro;
+clear H1;
+rewrite < plus_assoc in H;
+rewrite > opp_inverse in H;
+rewrite > zero_neutral in H;
+assumption.
+qed.
+
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