CoRN.model.setoids.Zsetoid
Lemma ap_Z_irreflexive : irreflexive (A:=Z) ap_Z.
Proof.
red in |- ×.
apply ap_Z_irreflexive0.
Qed.
Lemma ap_Z_symmetric : Csymmetric ap_Z.
Proof.
red in |- ×.
apply ap_Z_symmetric0.
Qed.
Lemma ap_Z_cotransitive : cotransitive (A:=Z) ap_Z.
Proof.
red in |- ×.
apply ap_Z_cotransitive0.
Qed.
Lemma ap_Z_tight : tight_apart (A:=Z) (eq (A:=Z)) ap_Z.
Proof.
red in |- ×.
apply ap_Z_tight0.
Qed.
Definition ap_Z_is_apartness := Build_is_CSetoid Z (eq (A:=Z)) ap_Z
ap_Z_irreflexive ap_Z_symmetric ap_Z_cotransitive ap_Z_tight.
Definition Z_as_CSetoid := Build_CSetoid _ _ _ ap_Z_is_apartness.
Canonical Structure Z_as_CSetoid.
The term Z_as_CSetoid is of type CSetoid. Hence we have proven that Z is a constructive setoid.
Addition
We will prove now that the addition on the integers is a setoid function.Lemma Zplus_wd :
bin_fun_wd Z_as_CSetoid Z_as_CSetoid Z_as_CSetoid Zplus.
Proof.
red in |- ×.
simpl in |- ×.
apply Zplus_wd0.
Qed.
Lemma Zplus_strext :
bin_fun_strext Z_as_CSetoid Z_as_CSetoid Z_as_CSetoid Zplus.
Proof.
red in |- ×.
simpl in |- ×.
apply Zplus_strext0.
Qed.
Definition Zplus_is_bin_fun := Build_CSetoid_bin_fun
Z_as_CSetoid Z_as_CSetoid Z_as_CSetoid Zplus Zplus_strext.
Canonical Structure Zplus_is_bin_fun.
What's more: the addition is also associative and commutative.
Lemma Zplus_is_assoc : associative Zplus_is_bin_fun.
Proof.
red in |- ×.
intros x y z.
simpl in |- ×.
apply Zplus_assoc.
Qed.
Lemma Zplus_is_commut : commutes Zplus_is_bin_fun.
Proof.
red in |- ×.
simpl in |- ×.
intros x y.
apply Zplus_comm.
Qed.
Lemma Zopp_wd : fun_wd (S1:=Z_as_CSetoid) (S2:=Z_as_CSetoid) Zopp.
Proof.
red in |- ×.
simpl in |- ×.
intros x y H.
apply (f_equal Zopp H).
Qed.
Lemma Zopp_strext :
fun_strext (S1:=Z_as_CSetoid) (S2:=Z_as_CSetoid) Zopp.
Proof.
red in |- ×.
simpl in |- ×.
unfold ap_Z in |- ×.
intros x y H.
intro H0.
apply H.
exact (f_equal Zopp H0).
Qed.
Definition Zopp_is_fun :=
Build_CSetoid_fun Z_as_CSetoid Z_as_CSetoid Zopp Zopp_strext.
Canonical Structure Zopp_is_fun.
Lemma Zmult_wd :
bin_fun_wd Z_as_CSetoid Z_as_CSetoid Z_as_CSetoid Zmult.
Proof.
red in |- ×.
simpl in |- ×.
intros x1 x2 y1 y2 H H0.
apply (f_equal2 Zmult (x1:=x1) (y1:=x2) (x2:=y1) (y2:=y2)).
assumption.
assumption.
Qed.
Lemma Zmult_strext :
bin_fun_strext Z_as_CSetoid Z_as_CSetoid Z_as_CSetoid Zmult.
Proof.
red in |- ×.
simpl in |- ×.
apply Zmult_strext0.
Qed.
Definition Zmult_is_bin_fun := Build_CSetoid_bin_fun
Z_as_CSetoid Z_as_CSetoid Z_as_CSetoid Zmult Zmult_strext.
Canonical Structure Zmult_is_bin_fun.
Lemma Zmult_is_assoc : associative Zmult_is_bin_fun.
Proof.
red in |- ×.
intros x y z.
simpl in |- ×.
apply Zmult_assoc.
Qed.
Lemma Zmult_is_commut : commutes Zmult_is_bin_fun.
Proof.
red in |- ×.
simpl in |- ×.
intros x y.
apply Zmult_comm.
Qed.
Lemma is_nullary_operation_Z_0 : (is_nullary_operation Z_as_CSetoid 0%Z).
Proof.
unfold is_nullary_operation.
intuition.
Qed.