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Require Import Raxioms.
Require Import Rdefinitions.
Require Export Ring Field.
Open Local Scope R_scope.
Lemma RTheory : ring_theory 0 1 Rplus Rmult Rminus Ropp (eq (A:=R)).
Proof.
constructor.
intro; apply Rplus_0_l.
exact Rplus_comm.
symmetry in |- *; apply Rplus_assoc.
intro; apply Rmult_1_l.
exact Rmult_comm.
symmetry in |- *; apply Rmult_assoc.
intros m n p.
rewrite Rmult_comm in |- *.
rewrite (Rmult_comm n p) in |- *.
rewrite (Rmult_comm m p) in |- *.
apply Rmult_plus_distr_l.
reflexivity.
exact Rplus_opp_r.
Qed.
Lemma Rfield : field_theory 0 1 Rplus Rmult Rminus Ropp Rdiv Rinv (eq(A:=R)).
Proof.
constructor.
exact RTheory.
exact R1_neq_R0.
reflexivity.
exact Rinv_l.
Qed.
Lemma Rlt_n_Sn : forall x, x < x + 1.
Proof.
intro.
elim archimed with x; intros.
destruct H0.
apply Rlt_trans with (IZR (up x)); trivial.
replace (IZR (up x)) with (x + (IZR (up x) - x))%R.
apply Rplus_lt_compat_l; trivial.
unfold Rminus in |- *.
rewrite (Rplus_comm (IZR (up x)) (- x)) in |- *.
rewrite <- Rplus_assoc in |- *.
rewrite Rplus_opp_r in |- *.
apply Rplus_0_l.
elim H0.
unfold Rminus in |- *.
rewrite (Rplus_comm (IZR (up x)) (- x)) in |- *.
rewrite <- Rplus_assoc in |- *.
rewrite Rplus_opp_r in |- *.
rewrite Rplus_0_l in |- *; trivial.
Qed.
Notation Rset := (Eqsth R).
Notation Rext := (Eq_ext Rplus Rmult Ropp).
Lemma Rlt_0_2 : 0 < 2.
apply Rlt_trans with (0 + 1).
apply Rlt_n_Sn.
rewrite Rplus_comm in |- *.
apply Rplus_lt_compat_l.
replace 1 with (0 + 1).
apply Rlt_n_Sn.
apply Rplus_0_l.
Qed.
Lemma Rgen_phiPOS : forall x, InitialRing.gen_phiPOS1 1 Rplus Rmult x > 0.
unfold Rgt in |- *.
induction x; simpl in |- *; intros.
apply Rlt_trans with (1 + 0).
rewrite Rplus_comm in |- *.
apply Rlt_n_Sn.
apply Rplus_lt_compat_l.
rewrite <- (Rmul_0_l Rset Rext RTheory 2) in |- *.
rewrite Rmult_comm in |- *.
apply Rmult_lt_compat_l.
apply Rlt_0_2.
trivial.
rewrite <- (Rmul_0_l Rset Rext RTheory 2) in |- *.
rewrite Rmult_comm in |- *.
apply Rmult_lt_compat_l.
apply Rlt_0_2.
trivial.
replace 1 with (0 + 1).
apply Rlt_n_Sn.
apply Rplus_0_l.
Qed.
Lemma Rgen_phiPOS_not_0 :
forall x, InitialRing.gen_phiPOS1 1 Rplus Rmult x <> 0.
red in |- *; intros.
specialize (Rgen_phiPOS x).
rewrite H in |- *; intro.
apply (Rlt_asym 0 0); trivial.
Qed.
Lemma Zeq_bool_complete : forall x y,
InitialRing.gen_phiZ 0%R 1%R Rplus Rmult Ropp x =
InitialRing.gen_phiZ 0%R 1%R Rplus Rmult Ropp y ->
Zeq_bool x y = true.
Proof gen_phiZ_complete Rset Rext Rfield Rgen_phiPOS_not_0.
Add Field RField : Rfield (infinite Zeq_bool_complete).
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