path: root/src/Algebra.v

Require Import Coq.Classes.Morphisms. Require Coq.Setoids.Setoid.
Require Import Crypto.Util.Tactics Crypto.Tactics.Nsatz.
Local Close Scope nat_scope. Local Close Scope type_scope. Local Close Scope core_scope.

Section Algebra.
  Context {T:Type} {eq:T->T->Prop}.
  Local Infix "=" := eq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.

  Class is_eq_dec := { eq_dec : forall x y : T, {x=y} + {x<>y} }.

  Section SingleOperation.
    Context {op:T->T->T}.

    Class is_associative := { associative : forall x y z, op x (op y z) = op (op x y) z }.

    Context {id:T}.

    Class is_left_identity := { left_identity : forall x, op id x = x }.
    Class is_right_identity := { right_identity : forall x, op x id = x }.

    Class monoid :=
      {
        monoid_is_associative : is_associative;
        monoid_is_left_identity : is_left_identity;
        monoid_is_right_identity : is_right_identity;

        monoid_op_Proper: Proper (respectful eq (respectful eq eq)) op;
        monoid_Equivalence : Equivalence eq;
        monoid_is_eq_dec : is_eq_dec
      }.
    Global Existing Instance monoid_is_associative.
    Global Existing Instance monoid_is_left_identity.
    Global Existing Instance monoid_is_right_identity.
    Global Existing Instance monoid_Equivalence.
    Global Existing Instance monoid_is_eq_dec.
    Global Existing Instance monoid_op_Proper.

    Context {inv:T->T}.
    Class is_left_inverse := { left_inverse : forall x, op (inv x) x = id }.
    Class is_right_inverse := { right_inverse : forall x, op x (inv x) = id }.

    Class group :=
      {
        group_monoid : monoid;
        group_is_left_inverse : is_left_inverse;
        group_is_right_inverse : is_right_inverse;

        group_inv_Proper: Proper (respectful eq eq) inv
      }.
    Global Existing Instance group_monoid.
    Global Existing Instance group_is_left_inverse.
    Global Existing Instance group_is_right_inverse.
    Global Existing Instance group_inv_Proper.

    Class is_commutative := { commutative : forall x y, op x y = op y x }.

    Record abelian_group :=
      {
        abelian_group_group : group;
        abelian_group_is_commutative : is_commutative
      }.
    Existing Class abelian_group.
    Global Existing Instance abelian_group_group.
    Global Existing Instance abelian_group_is_commutative.
  End SingleOperation.

  Section AddMul.
    Context {zero one:T}. Local Notation "0" := zero. Local Notation "1" := one.
    Context {opp:T->T}. Local Notation "- x" := (opp x).
    Context {add:T->T->T} {sub:T->T->T} {mul:T->T->T}.
    Local Infix "+" := add. Local Infix "-" := sub. Local Infix "*" := mul.

    Class is_left_distributive := { left_distributive : forall a b c, a * (b + c) =  a * b + a * c }.
    Class is_right_distributive := { right_distributive : forall a b c, (b + c) * a = b * a + c * a }.


    Class ring :=
      {
        ring_abelian_group_add : abelian_group (op:=add) (id:=zero) (inv:=opp);
        ring_monoid_mul : monoid (op:=mul) (id:=one);
        ring_is_left_distributive : is_left_distributive;
        ring_is_right_distributive : is_right_distributive;

        ring_sub_definition : forall x y, x - y = x + opp y;

        ring_mul_Proper : Proper (respectful eq (respectful eq eq)) mul;
        ring_sub_Proper : Proper(respectful eq (respectful eq eq)) sub
      }.
    Global Existing Instance ring_abelian_group_add.
    Global Existing Instance ring_monoid_mul.
    Global Existing Instance ring_is_left_distributive.
    Global Existing Instance ring_is_right_distributive.
    Global Existing Instance ring_mul_Proper.
    Global Existing Instance ring_sub_Proper.

    Class commutative_ring :=
      {
        commutative_ring_ring : ring;
        commutative_ring_is_commutative : is_commutative (op:=mul)
      }.
    Global Existing Instance commutative_ring_ring.
    Global Existing Instance commutative_ring_is_commutative.

    Class is_mul_nonzero_nonzero := { mul_nonzero_nonzero : forall x y, x<>0 -> y<>0 -> x*y<>0 }.

    Class is_zero_neq_one := { zero_neq_one : zero <> one }.

    Class integral_domain :=
      {
        integral_domain_commutative_ring : commutative_ring;
        integral_domain_is_mul_nonzero_nonzero : is_mul_nonzero_nonzero;
        integral_domain_is_zero_neq_one : is_zero_neq_one
      }.
    Global Existing Instance integral_domain_commutative_ring.
    Global Existing Instance integral_domain_is_mul_nonzero_nonzero.
    Global Existing Instance integral_domain_is_zero_neq_one.

    Context {inv:T->T} {div:T->T->T}.
    Class is_left_multiplicative_inverse := { left_multiplicative_inverse : forall x, x<>0 -> (inv x) * x = 1 }.

    Class field :=
      {
        field_commutative_ring : commutative_ring;
        field_is_left_multiplicative_inverse : is_left_multiplicative_inverse;
        field_domain_is_zero_neq_one : is_zero_neq_one;

        field_div_definition : forall x y , div x y = x * inv y;

        field_inv_Proper : Proper (respectful eq eq) inv;
        field_div_Proper : Proper (respectful eq (respectful eq eq)) div
      }.
    Global Existing Instance field_commutative_ring.
    Global Existing Instance field_is_left_multiplicative_inverse.
    Global Existing Instance field_domain_is_zero_neq_one.
    Global Existing Instance field_inv_Proper.
    Global Existing Instance field_div_Proper.
  End AddMul.
End Algebra.


Module Monoid.
  Section Monoid.
    Context {T eq op id} {monoid:@monoid T eq op id}.
    Local Infix "=" := eq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
    Local Infix "*" := op.
    Local Infix "=" := eq : eq_scope.
    Local Open Scope eq_scope.

    Lemma cancel_right z iz (Hinv:op z iz = id) :
      forall x y, x * z = y * z <-> x = y.
    Proof.
      split; intros.
      { assert (op (op x z) iz = op (op y z) iz) as Hcut by (f_equiv; assumption).
        rewrite <-associative in Hcut.
        rewrite <-!associative, !Hinv, !right_identity in Hcut; exact Hcut. }
      { f_equiv; assumption. }
    Qed.

    Lemma cancel_left z iz (Hinv:op iz z = id) :
      forall x y, z * x = z * y <-> x = y.
    Proof.
      split; intros.
      { assert (op iz (op z x) = op iz (op z y)) as Hcut by (f_equiv; assumption).
        rewrite !associative, !Hinv, !left_identity in Hcut; exact Hcut. }
      { f_equiv; assumption. }
    Qed.

    Lemma inv_inv x ix iix : ix*x = id -> iix*ix = id -> iix = x.
    Proof.
      intros Hi Hii.
      assert (H:op iix id = op iix (op ix x)) by (rewrite Hi; reflexivity).
      rewrite associative, Hii, left_identity, right_identity in H; exact H.
    Qed.

    Lemma inv_op x y ix iy : ix*x = id -> iy*y = id -> (iy*ix)*(x*y) =id.
    Proof.
      intros Hx Hy.
      cut (iy * (ix*x) * y = id); try intro H.
      { rewrite <-!associative; rewrite <-!associative in H; exact H. }
      rewrite Hx, right_identity, Hy. reflexivity.
    Qed.

  End Monoid.
End Monoid.

Section ZeroNeqOne.
  Context {T eq zero one} `{@is_zero_neq_one T eq zero one} `{Equivalence T eq}.

  Lemma one_neq_zero : not (eq one zero).
  Proof.
    intro HH; symmetry in HH. auto using zero_neq_one.
  Qed.
End ZeroNeqOne.

Module Group.
  Section BasicProperties.
    Context {T eq op id inv} `{@group T eq op id inv}.
    Local Infix "=" := eq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
    Local Infix "*" := op.
    Local Infix "=" := eq : eq_scope.
    Local Open Scope eq_scope.

    Lemma cancel_left : forall z x y, z*x = z*y <-> x = y.
    Proof. eauto using Monoid.cancel_left, left_inverse. Qed.
    Lemma cancel_right : forall z x y, x*z = y*z <-> x = y.
    Proof. eauto using Monoid.cancel_right, right_inverse. Qed.
    Lemma inv_inv x : inv(inv(x)) = x.
    Proof. eauto using Monoid.inv_inv, left_inverse. Qed.
    Lemma inv_op x y : (inv y*inv x)*(x*y) =id.
    Proof. eauto using Monoid.inv_op, left_inverse. Qed.

    Lemma inv_unique x ix : ix * x = id -> ix = inv x.
    Proof.
      intro Hix.
      cut (ix*x*inv x = inv x).
      - rewrite <-associative, right_inverse, right_identity; trivial.
      - rewrite Hix, left_identity; reflexivity.
    Qed.

    Lemma inv_id : inv id = id.
    Proof. symmetry. eapply inv_unique, left_identity. Qed.

    Lemma inv_nonzero_nonzero : forall x, x <> id -> inv x <> id.
    Proof.
      intros ? Hx Ho.
      assert (Hxo: x * inv x = id) by (rewrite right_inverse; reflexivity).
      rewrite Ho, right_identity in Hxo. intuition.
   Qed.

    Section ZeroNeqOne.
      Context {one} `{is_zero_neq_one T eq id one}.
      Lemma opp_one_neq_zero : inv one <> id.
      Proof. apply inv_nonzero_nonzero, one_neq_zero. Qed.
      Lemma zero_neq_opp_one : id <> inv one.
      Proof. intro Hx. symmetry in Hx. eauto using opp_one_neq_zero. Qed.
    End ZeroNeqOne.
  End BasicProperties.

  Section Homomorphism.
    Context {G EQ OP ID INV} {groupG:@group G EQ OP ID INV}.
    Context {H eq op id inv} {groupH:@group H eq op id inv}.
    Context {phi:G->H}.
    Local Infix "=" := eq. Local Infix "=" := eq : type_scope.

    Class is_homomorphism :=
      {
        homomorphism : forall a b,  phi (OP a b) = op (phi a) (phi b);

        is_homomorphism_phi_proper : Proper (respectful EQ eq) phi
      }.
    Global Existing Instance is_homomorphism_phi_proper.
    Context `{is_homomorphism}.

    Lemma homomorphism_id : phi ID = id.
    Proof.
      assert (Hii: op (phi ID) (phi ID) = op (phi ID) id) by
        (rewrite <- homomorphism, left_identity, right_identity; reflexivity).
      rewrite cancel_left in Hii; exact Hii.
    Qed.

    Lemma homomorphism_inv : forall x, phi (INV x) = inv (phi x).
    Proof.
    Admitted.
  End Homomorphism.
End Group.

Require Coq.nsatz.Nsatz.

Ltac dropAlgebraSyntax :=
  cbv beta delta [
      Algebra_syntax.zero
        Algebra_syntax.one
        Algebra_syntax.addition
        Algebra_syntax.multiplication
        Algebra_syntax.subtraction
        Algebra_syntax.opposite
        Algebra_syntax.equality
        Algebra_syntax.bracket
        Algebra_syntax.power
        ] in *.

Ltac dropRingSyntax :=
  dropAlgebraSyntax;
  cbv beta delta [
      Ncring.zero_notation
        Ncring.one_notation
        Ncring.add_notation
        Ncring.mul_notation
        Ncring.sub_notation
        Ncring.opp_notation
        Ncring.eq_notation
    ] in *.

Module Ring.
  Section Ring.
    Context {T eq zero one opp add sub mul} `{@ring T eq zero one opp add sub mul}.
    Local Infix "=" := eq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
    Local Notation "0" := zero. Local Notation "1" := one.
    Local Infix "+" := add. Local Infix "-" := sub. Local Infix "*" := mul.

    Lemma mul_0_r : forall x, 0 * x = 0.
    Proof.
      intros.
      assert (0*x = 0*x) as Hx by reflexivity.
      rewrite <-(right_identity 0), right_distributive in Hx at 1.
      assert (0*x + 0*x - 0*x = 0*x - 0*x) as Hxx by (f_equiv; exact Hx).
      rewrite !ring_sub_definition, <-associative, right_inverse, right_identity in Hxx; exact Hxx.
    Qed.

    Lemma mul_0_l : forall x, x * 0 = 0.
    Proof.
      intros.
      assert (x*0 = x*0) as Hx by reflexivity.
      rewrite <-(left_identity 0), left_distributive in Hx at 1.
      assert (opp (x*0) + (x*0 + x*0)  = opp (x*0) + x*0) as Hxx by (f_equiv; exact Hx).
      rewrite associative, left_inverse, left_identity in Hxx; exact Hxx.
    Qed.

    Lemma sub_0_l x : 0 - x = opp x.
    Proof. rewrite ring_sub_definition. rewrite left_identity. reflexivity. Qed.

    Lemma mul_opp_r x y : x * opp y = opp (x * y).
    Proof.
      assert (Ho:x*(opp y) + x*y = 0)
        by (rewrite <-left_distributive, left_inverse, mul_0_l; reflexivity).
      rewrite <-(left_identity (opp (x*y))), <-Ho; clear Ho.
      rewrite <-!associative, right_inverse, right_identity; reflexivity.
    Qed.

    Lemma mul_opp_l x y : opp x * y = opp (x * y).
    Proof.
      assert (Ho:opp x*y + x*y = 0)
        by (rewrite <-right_distributive, left_inverse, mul_0_r; reflexivity).
      rewrite <-(left_identity (opp (x*y))), <-Ho; clear Ho.
      rewrite <-!associative, right_inverse, right_identity; reflexivity.
    Qed.

    Definition opp_nonzero_nonzero : forall x, x <> 0 -> opp x <> 0 := Group.inv_nonzero_nonzero.

    Global Instance is_left_distributive_sub : is_left_distributive (eq:=eq)(add:=sub)(mul:=mul).
    Proof.
      split; intros. rewrite !ring_sub_definition, left_distributive.
      eapply Group.cancel_left, mul_opp_r.
    Qed.

    Global Instance is_right_distributive_sub : is_right_distributive (eq:=eq)(add:=sub)(mul:=mul).
    Proof.
      split; intros. rewrite !ring_sub_definition, right_distributive.
      eapply Group.cancel_left, mul_opp_l.
    Qed.

    Global Instance Ncring_Ring_ops : @Ncring.Ring_ops T zero one add mul sub opp eq.
    Global Instance Ncring_Ring : @Ncring.Ring T zero one add mul sub opp eq Ncring_Ring_ops.
    Proof.
      split; dropRingSyntax; eauto using left_identity, right_identity, commutative, associative, right_inverse, left_distributive, right_distributive, ring_sub_definition with core typeclass_instances.
      - (* TODO: why does [eauto using @left_identity with typeclass_instances] not work? *)
        eapply @left_identity; eauto with typeclass_instances.
      - eapply @right_identity; eauto with typeclass_instances.
      - eapply associative.
      - intros; eapply right_distributive.
      - intros; eapply left_distributive.
    Qed.
  End Ring.

  Section Homomorphism.
    Context {R EQ ZERO ONE OPP ADD SUB MUL} `{@ring R EQ ZERO ONE OPP ADD SUB MUL}.
    Context {S eq zero one opp add sub mul} `{@ring S eq zero one opp add sub mul}.
    Context {phi:R->S}.
    Local Infix "=" := eq. Local Infix "=" := eq : type_scope.

    Class is_homomorphism :=
      {
        homomorphism_is_homomorphism : Group.is_homomorphism (phi:=phi) (OP:=ADD) (op:=add) (EQ:=EQ) (eq:=eq);
        homomorphism_mul : forall x y, phi (MUL x y) = mul (phi x) (phi y);
        homomorphism_one : phi ONE = one
      }.
    Global Existing Instance homomorphism_is_homomorphism.

    Context `{is_homomorphism}.

    Lemma homomorphism_add : forall x y,  phi (ADD x y) = add (phi x) (phi y).
    Proof. apply Group.homomorphism. Qed.

    Definition homomorphism_opp : forall x,  phi (OPP x) = opp (phi x) :=
      (Group.homomorphism_inv (INV:=OPP) (inv:=opp)).

    Lemma homomorphism_sub : forall x y, phi (SUB x y) = sub (phi x) (phi y).
    Proof.
      intros.
      rewrite !ring_sub_definition, Group.homomorphism, homomorphism_opp. reflexivity.
    Qed.

  End Homomorphism.

  Section TacticSupportCommutative.
    Context {T eq zero one opp add sub mul} `{@commutative_ring T eq zero one opp add sub mul}.

    Global Instance Cring_Cring_commutative_ring :
      @Cring.Cring T zero one add mul sub opp eq Ring.Ncring_Ring_ops Ring.Ncring_Ring.
    Proof. unfold Cring.Cring; intros; dropRingSyntax. eapply commutative. Qed.

   Lemma ring_theory_for_stdlib_tactic : Ring_theory.ring_theory zero one add mul sub opp eq.
   Proof.
     constructor; intros. (* TODO(automation): make [auto] do this? *)
     - apply left_identity.
     - apply commutative.
     - apply associative.
     - apply left_identity.
     - apply commutative.
     - apply associative.
     - apply right_distributive.
     - apply ring_sub_definition.
     - apply right_inverse.
   Qed.
  End TacticSupportCommutative.
End Ring.

Module IntegralDomain.
  Section IntegralDomain.
    Context {T eq zero one opp add sub mul} `{@integral_domain T eq zero one opp add sub mul}.

    Lemma mul_nonzero_nonzero_cases (x y : T)
      : eq (mul x  y) zero -> eq x zero \/ eq y zero.
    Proof.
      pose proof mul_nonzero_nonzero x y.
      destruct (eq_dec x zero); destruct (eq_dec y zero); intuition.
    Qed.

    Global Instance Integral_domain :
      @Integral_domain.Integral_domain T zero one add mul sub opp eq Ring.Ncring_Ring_ops
                                       Ring.Ncring_Ring Ring.Cring_Cring_commutative_ring.
    Proof.
      split; dropRingSyntax.
      - auto using mul_nonzero_nonzero_cases.
      - intro bad; symmetry in bad; auto using zero_neq_one.
    Qed.
  End IntegralDomain.
End IntegralDomain.

Module Field.
  Section Field.
    Context {T eq zero one opp add mul sub inv div} `{@field T eq zero one opp add sub mul inv div}.
    Local Infix "=" := eq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
    Local Notation "0" := zero. Local Notation "1" := one.
    Local Infix "+" := add. Local Infix "*" := mul.

    Global Instance is_mul_nonzero_nonzero : @is_mul_nonzero_nonzero T eq 0 mul.
    Proof.
      constructor. intros x y Hx Hy Hxy.
      assert (0 = (inv y * (inv x * x)) * y) as H00. (rewrite <-!associative, Hxy, !Ring.mul_0_l; reflexivity).
      rewrite left_multiplicative_inverse in H00 by assumption.
      rewrite right_identity in H00.
      rewrite left_multiplicative_inverse in H00 by assumption.
      auto using zero_neq_one.
    Qed.

    Global Instance integral_domain : @integral_domain T eq zero one opp add sub mul.
    Proof.
      split; auto using field_commutative_ring, field_domain_is_zero_neq_one, is_mul_nonzero_nonzero.
    Qed.

    Require Coq.setoid_ring.Field_theory.
    Lemma field_theory_for_stdlib_tactic : Field_theory.field_theory 0 1 add mul sub opp div inv eq.
    Proof.
      constructor.
      { apply Ring.ring_theory_for_stdlib_tactic. }
      { intro H01. symmetry in H01. auto using zero_neq_one. }
      { apply field_div_definition. }
      { apply left_multiplicative_inverse. }
    Qed.

  End Field.

  Section Homomorphism.
    Context {F EQ ZERO ONE OPP ADD MUL SUB INV DIV} `{@field F EQ ZERO ONE OPP ADD SUB MUL INV DIV}.
    Context {K eq zero one opp add mul sub inv div} `{@field K eq zero one opp add sub mul inv div}.
    Context {phi:F->K}.
    Local Infix "=" := eq. Local Infix "=" := eq : type_scope.
    Context `{@Ring.is_homomorphism F EQ ONE ADD MUL K eq one add mul phi}.

    Lemma homomorphism_multiplicative_inverse : forall x, phi (INV x) = inv (phi x). Admitted.

    Lemma homomorphism_div : forall x y, phi (DIV x y) = div (phi x) (phi y).
    Proof.
      intros. rewrite !field_div_definition.
      rewrite Ring.homomorphism_mul, homomorphism_multiplicative_inverse. reflexivity.
    Qed.
  End Homomorphism.
End Field.

(*** Tactics for manipulating field equations *)
Require Import Coq.setoid_ring.Field_tac.

Ltac guess_field :=
  match goal with
  | |- ?eq _ _ =>  constr:(_:field (eq:=eq))
  | |- not (?eq _ _) =>  constr:(_:field (eq:=eq))
  | [H: ?eq _ _ |- _ ] =>  constr:(_:field (eq:=eq))
  | [H: not (?eq _ _) |- _] =>  constr:(_:field (eq:=eq))
  end.

Ltac common_denominator :=
  let fld := guess_field in
  lazymatch type of fld with
    field (div:=?div) =>
    lazymatch goal with
    | |- appcontext[div] => field_simplify_eq
    | |- _ => idtac
    end
  end.

Ltac common_denominator_in H :=
  let fld := guess_field in
  lazymatch type of fld with
    field (div:=?div) =>
    lazymatch type of H with
    | appcontext[div] => field_simplify_eq in H
    | _ => idtac
    end
  end.

Ltac common_denominator_all :=
  common_denominator;
  repeat match goal with [H: _ |- _ _ _ ] => progress common_denominator_in H end.

Inductive field_simplify_done {T} : T -> Type :=
  Field_simplify_done : forall H, field_simplify_done H.

Ltac field_simplify_eq_hyps :=
  repeat match goal with
           [ H: _ |- _ ] =>
           match goal with
           | [ Ha : field_simplify_done H |- _ ] => fail
           | _ => idtac
           end;
           field_simplify_eq in H;
           unique pose proof (Field_simplify_done H)
         end;
  repeat match goal with [ H: field_simplify_done _ |- _] => clear H end.

Ltac field_simplify_eq_all := field_simplify_eq_hyps; try field_simplify_eq.


(*** Polynomial equations over fields *)

Ltac neq01 :=
  try solve
      [apply zero_neq_one
      |apply Group.zero_neq_opp_one
      |apply one_neq_zero
      |apply Group.opp_one_neq_zero].

Ltac field_algebra :=
  intros;
  common_denominator_all;
  try (nsatz; dropRingSyntax);
  repeat (apply conj);
  try solve
      [neq01
      |trivial
      |apply Ring.opp_nonzero_nonzero;trivial].

Section Example.
  Context {F zero one opp add sub mul inv div} `{F_field:field F eq zero one opp add sub mul inv div}.
  Local Infix "+" := add. Local Infix "*" := mul. Local Infix "-" := sub. Local Infix "/" := div.
  Local Notation "0" := zero. Local Notation "1" := one.

  Add Field _ExampleField : (Field.field_theory_for_stdlib_tactic (T:=F)).

  Example _example_nsatz x y : 1+1 <> 0 -> x + y = 0 -> x - y = 0 -> x = 0.
  Proof. field_algebra. Qed.

  Example _example_field_nsatz x y z : y <> 0 -> x/y = z -> z*y + y = x + y.
  Proof. intros; subst; field_algebra. Qed.

  Example _example_nonzero_nsatz_contradict x y : x * y = 1 -> not (x = 0).
  Proof. intros. intro. nsatz_contradict. Qed.
End Example.

Section Z.
  Require Import ZArith.
  Global Instance ring_Z : @ring Z Logic.eq 0%Z 1%Z Z.opp Z.add Z.sub Z.mul.
  Proof. repeat split; auto using Z.eq_dec with zarith typeclass_instances. Qed.

  Global Instance commutative_ring_Z : @commutative_ring Z Logic.eq 0%Z 1%Z Z.opp Z.add Z.sub Z.mul.
  Proof. eauto using @commutative_ring, @is_commutative, ring_Z with zarith. Qed.

  Global Instance integral_domain_Z : @integral_domain Z Logic.eq 0%Z 1%Z Z.opp Z.add Z.sub Z.mul.
  Proof.
    split.
    { apply commutative_ring_Z. }
    { constructor. intros. apply Z.neq_mul_0; auto. }
    { constructor. discriminate. }
  Qed.

  Example _example_nonzero_nsatz_contradict_Z x y : Z.mul x y = (Zpos xH) -> not (x = Z0).
  Proof. intros. intro. nsatz_contradict. Qed.
End Z.