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|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: R_Ifp.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*)
(**********************************************************)
(** Complements for the reals.Integer and fractional part *)
(* *)
(**********************************************************)
Require Rbase.
Require Omega.
V7only [ Import nat_scope. Import Z_scope. Import R_scope. ].
Open Local Scope R_scope.
(*********************************************************)
(** Fractional part *)
(*********************************************************)
(**********)
Definition Int_part:R->Z:=[r:R](`(up r)-1`).
(**********)
Definition frac_part:R->R:=[r:R](Rminus r (IZR (Int_part r))).
(**********)
Lemma tech_up:(r:R)(z:Z)(Rlt r (IZR z))->(Rle (IZR z) (Rplus r R1))->
z=(up r).
Intros;Generalize (archimed r);Intro;Elim H1;Intros;Clear H1;
Unfold Rgt in H2;Unfold Rminus in H3;
Generalize (Rle_compatibility r (Rplus (IZR (up r))
(Ropp r)) R1 H3);Intro;Clear H3;
Rewrite (Rplus_sym (IZR (up r)) (Ropp r)) in H1;
Rewrite <-(Rplus_assoc r (Ropp r) (IZR (up r))) in H1;
Rewrite (Rplus_Ropp_r r) in H1;Elim (Rplus_ne (IZR (up r)));Intros a b;
Rewrite b in H1;Clear a b;Apply (single_z_r_R1 r z (up r));Auto with zarith real.
Qed.
(**********)
Lemma up_tech:(r:R)(z:Z)(Rle (IZR z) r)->(Rlt r (IZR `z+1`))->
`z+1`=(up r).
Intros;Generalize (Rle_compatibility R1 (IZR z) r H);Intro;Clear H;
Rewrite (Rplus_sym R1 (IZR z)) in H1;Rewrite (Rplus_sym R1 r) in H1;
Cut (R1==(IZR `1`));Auto with zarith real.
Intro;Generalize H1;Pattern 1 R1;Rewrite H;Intro;Clear H H1;
Rewrite <-(plus_IZR z `1`) in H2;Apply (tech_up r `z+1`);Auto with zarith real.
Qed.
(**********)
Lemma fp_R0:(frac_part R0)==R0.
Unfold frac_part; Unfold Int_part; Elim (archimed R0);
Intros; Unfold Rminus;
Elim (Rplus_ne (Ropp (IZR `(up R0)-1`))); Intros a b;
Rewrite b;Clear a b;Rewrite <- Z_R_minus;Cut (up R0)=`1`.
Intro;Rewrite H1;
Rewrite (eq_Rminus (IZR `1`) (IZR `1`) (refl_eqT R (IZR `1`)));
Apply Ropp_O.
Elim (archimed R0);Intros;Clear H2;Unfold Rgt in H1;
Rewrite (minus_R0 (IZR (up R0))) in H0;
Generalize (lt_O_IZR (up R0) H1);Intro;Clear H1;
Generalize (le_IZR_R1 (up R0) H0);Intro;Clear H H0;Omega.
Qed.
(**********)
Lemma for_base_fp:(r:R)(Rgt (Rminus (IZR (up r)) r) R0)/\
(Rle (Rminus (IZR (up r)) r) R1).
Intro; Split;
Cut (Rgt (IZR (up r)) r)/\(Rle (Rminus (IZR (up r)) r) R1).
Intro; Elim H; Intros.
Apply (Rgt_minus (IZR (up r)) r H0).
Apply archimed.
Intro; Elim H; Intros.
Exact H1.
Apply archimed.
Qed.
(**********)
Lemma base_fp:(r:R)(Rge (frac_part r) R0)/\(Rlt (frac_part r) R1).
Intro; Unfold frac_part; Unfold Int_part; Split.
(*sup a O*)
Cut (Rge (Rminus r (IZR (up r))) (Ropp R1)).
Rewrite <- Z_R_minus;Simpl;Intro; Unfold Rminus;
Rewrite Ropp_distr1;Rewrite <-Rplus_assoc;
Fold (Rminus r (IZR (up r)));
Fold (Rminus (Rminus r (IZR (up r))) (Ropp R1));
Apply Rge_minus;Auto with zarith real.
Rewrite <- Ropp_distr2;Apply Rle_Ropp;Elim (for_base_fp r); Auto with zarith real.
(*inf a 1*)
Cut (Rlt (Rminus r (IZR (up r))) R0).
Rewrite <- Z_R_minus; Simpl;Intro; Unfold Rminus;
Rewrite Ropp_distr1;Rewrite <-Rplus_assoc;
Fold (Rminus r (IZR (up r)));Rewrite Ropp_Ropp;
Elim (Rplus_ne R1);Intros a b;Pattern 2 R1;Rewrite <-a;Clear a b;
Rewrite (Rplus_sym (Rminus r (IZR (up r))) R1);
Apply Rlt_compatibility;Auto with zarith real.
Elim (for_base_fp r);Intros;Rewrite <-Ropp_O;
Rewrite<-Ropp_distr2;Apply Rgt_Ropp;Auto with zarith real.
Qed.
(*********************************************************)
(** Properties *)
(*********************************************************)
(**********)
Lemma base_Int_part:(r:R)(Rle (IZR (Int_part r)) r)/\
(Rgt (Rminus (IZR (Int_part r)) r) (Ropp R1)).
Intro;Unfold Int_part;Elim (archimed r);Intros.
Split;Rewrite <- (Z_R_minus (up r) `1`);Simpl.
Generalize (Rle_minus (Rminus (IZR (up r)) r) R1 H0);Intro;
Unfold Rminus in H1;
Rewrite (Rplus_assoc (IZR (up r)) (Ropp r) (Ropp R1)) in
H1;Rewrite (Rplus_sym (Ropp r) (Ropp R1)) in H1;
Rewrite <-(Rplus_assoc (IZR (up r)) (Ropp R1) (Ropp r)) in
H1;Fold (Rminus (IZR (up r)) R1) in H1;
Fold (Rminus (Rminus (IZR (up r)) R1) r) in H1;
Apply Rminus_le;Auto with zarith real.
Generalize (Rgt_plus_plus_r (Ropp R1) (IZR (up r)) r H);Intro;
Rewrite (Rplus_sym (Ropp R1) (IZR (up r))) in H1;
Generalize (Rgt_plus_plus_r (Ropp r)
(Rplus (IZR (up r)) (Ropp R1)) (Rplus (Ropp R1) r) H1);
Intro;Clear H H0 H1;
Rewrite (Rplus_sym (Ropp r) (Rplus (IZR (up r)) (Ropp R1)))
in H2;Fold (Rminus (IZR (up r)) R1) in H2;
Fold (Rminus (Rminus (IZR (up r)) R1) r) in H2;
Rewrite (Rplus_sym (Ropp r) (Rplus (Ropp R1) r)) in H2;
Rewrite (Rplus_assoc (Ropp R1) r (Ropp r)) in H2;
Rewrite (Rplus_Ropp_r r) in H2;Elim (Rplus_ne (Ropp R1));Intros a b;
Rewrite a in H2;Clear a b;Auto with zarith real.
Qed.
(**********)
Lemma Int_part_INR:(n : nat) (Int_part (INR n)) = (inject_nat n).
Intros n; Unfold Int_part.
Cut (up (INR n)) = (Zplus (inject_nat n) (inject_nat (1))).
Intros H'; Rewrite H'; Simpl; Ring.
Apply sym_equal; Apply tech_up; Auto.
Replace (Zplus (inject_nat n) (inject_nat (1))) with (INZ (S n)).
Repeat Rewrite <- INR_IZR_INZ.
Apply lt_INR; Auto.
Rewrite Zplus_sym; Rewrite <- inj_plus; Simpl; Auto.
Rewrite plus_IZR; Simpl; Auto with real.
Repeat Rewrite <- INR_IZR_INZ; Auto with real.
Qed.
(**********)
Lemma fp_nat:(r:R)(frac_part r)==R0->(Ex [c:Z](r==(IZR c))).
Unfold frac_part;Intros;Split with (Int_part r);Apply Rminus_eq; Auto with zarith real.
Qed.
(**********)
Lemma R0_fp_O:(r:R)~R0==(frac_part r)->~R0==r.
Red;Intros;Rewrite <- H0 in H;Generalize fp_R0;Intro;Auto with zarith real.
Qed.
(**********)
Lemma Rminus_Int_part1:(r1,r2:R)(Rge (frac_part r1) (frac_part r2))->
(Int_part (Rminus r1 r2))=(Zminus (Int_part r1) (Int_part r2)).
Intros;Elim (base_fp r1);Elim (base_fp r2);Intros;
Generalize (Rle_sym2 R0 (frac_part r2) H0);Intro;Clear H0;
Generalize (Rle_Ropp R0 (frac_part r2) H4);Intro;Clear H4;
Rewrite (Ropp_O) in H0;
Generalize (Rle_sym2 (Ropp (frac_part r2)) R0 H0);Intro;Clear H0;
Generalize (Rle_sym2 R0 (frac_part r1) H2);Intro;Clear H2;
Generalize (Rlt_Ropp (frac_part r2) R1 H1);Intro;Clear H1;
Unfold Rgt in H2;
Generalize (sum_inequa_Rle_lt R0 (frac_part r1) R1 (Ropp R1)
(Ropp (frac_part r2)) R0 H0 H3 H2 H4);Intro;Elim H1;Intros;
Clear H1;Elim (Rplus_ne R1);Intros a b;Rewrite a in H6;Clear a b H5;
Generalize (Rge_minus (frac_part r1) (frac_part r2) H);Intro;Clear H;
Fold (Rminus (frac_part r1) (frac_part r2)) in H6;
Generalize (Rle_sym2 R0 (Rminus (frac_part r1) (frac_part r2)) H1);
Intro;Clear H1 H3 H4 H0 H2;Unfold frac_part in H6 H;
Unfold Rminus in H6 H;
Rewrite (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))) in H;
Rewrite (Ropp_Ropp (IZR (Int_part r2))) in H;
Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1)))
(Rplus (Ropp r2) (IZR (Int_part r2)))) in H;
Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2)
(IZR (Int_part r2))) in H;
Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (Ropp r2)) in H;
Rewrite (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1)))
(IZR (Int_part r2))) in H;
Rewrite <-(Rplus_assoc r1 (Ropp r2)
(Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))) in H;
Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) in H;
Fold (Rminus r1 r2) in H;Fold (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))
in H;Generalize (Rle_compatibility
(Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) R0
(Rplus (Rminus r1 r2) (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) H);Intro;
Clear H;Rewrite (Rplus_sym (Rminus r1 r2)
(Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) in H0;
Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) (Rminus r1 r2)) in H0;
Unfold Rminus in H0;Fold (Rminus r1 r2) in H0;
Rewrite (Rplus_assoc (IZR (Int_part r1)) (Ropp (IZR (Int_part r2)))
(Rplus (IZR (Int_part r2)) (Ropp (IZR (Int_part r1))))) in H0;
Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r2))) (IZR (Int_part r2))
(Ropp (IZR (Int_part r1)))) in H0;Rewrite (Rplus_Ropp_l (IZR (Int_part r2))) in
H0;Elim (Rplus_ne (Ropp (IZR (Int_part r1))));Intros a b;Rewrite b in H0;
Clear a b;
Elim (Rplus_ne (Rplus (IZR (Int_part r1)) (Ropp (IZR (Int_part r2)))));
Intros a b;Rewrite a in H0;Clear a b;Rewrite (Rplus_Ropp_r (IZR (Int_part r1)))
in H0;Elim (Rplus_ne (Rminus r1 r2));Intros a b;Rewrite b in H0;
Clear a b;Fold (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) in H0;
Rewrite (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))) in H6;
Rewrite (Ropp_Ropp (IZR (Int_part r2))) in H6;
Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1)))
(Rplus (Ropp r2) (IZR (Int_part r2)))) in H6;
Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2)
(IZR (Int_part r2))) in H6;
Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (Ropp r2)) in H6;
Rewrite (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1)))
(IZR (Int_part r2))) in H6;
Rewrite <-(Rplus_assoc r1 (Ropp r2)
(Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))) in H6;
Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) in H6;
Fold (Rminus r1 r2) in H6;Fold (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))
in H6;Generalize (Rlt_compatibility
(Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Rplus (Rminus r1 r2) (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) R1 H6);
Intro;Clear H6;
Rewrite (Rplus_sym (Rminus r1 r2)
(Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) in H;
Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) (Rminus r1 r2)) in H;
Rewrite <-(Ropp_distr2 (IZR (Int_part r1)) (IZR (Int_part r2))) in H;
Rewrite (Rplus_Ropp_r (Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H;
Elim (Rplus_ne (Rminus r1 r2));Intros a b;Rewrite b in H;Clear a b;
Rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0;
Rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H;
Cut R1==(IZR `1`);Auto with zarith real.
Intro;Rewrite H1 in H;Clear H1;
Rewrite <-(plus_IZR `(Int_part r1)-(Int_part r2)` `1`) in H;
Generalize (up_tech (Rminus r1 r2) `(Int_part r1)-(Int_part r2)`
H0 H);Intros;Clear H H0;Unfold 1 Int_part;Omega.
Qed.
(**********)
Lemma Rminus_Int_part2:(r1,r2:R)(Rlt (frac_part r1) (frac_part r2))->
(Int_part (Rminus r1 r2))=(Zminus (Zminus (Int_part r1) (Int_part r2)) `1`).
Intros;Elim (base_fp r1);Elim (base_fp r2);Intros;
Generalize (Rle_sym2 R0 (frac_part r2) H0);Intro;Clear H0;
Generalize (Rle_Ropp R0 (frac_part r2) H4);Intro;Clear H4;
Rewrite (Ropp_O) in H0;
Generalize (Rle_sym2 (Ropp (frac_part r2)) R0 H0);Intro;Clear H0;
Generalize (Rle_sym2 R0 (frac_part r1) H2);Intro;Clear H2;
Generalize (Rlt_Ropp (frac_part r2) R1 H1);Intro;Clear H1;
Unfold Rgt in H2;
Generalize (sum_inequa_Rle_lt R0 (frac_part r1) R1 (Ropp R1)
(Ropp (frac_part r2)) R0 H0 H3 H2 H4);Intro;Elim H1;Intros;
Clear H1;Elim (Rplus_ne (Ropp R1));Intros a b;Rewrite b in H5;
Clear a b H6;Generalize (Rlt_minus (frac_part r1) (frac_part r2) H);
Intro;Clear H;Fold (Rminus (frac_part r1) (frac_part r2)) in H5;
Clear H3 H4 H0 H2;Unfold frac_part in H5 H1;
Unfold Rminus in H5 H1;
Rewrite (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))) in H5;
Rewrite (Ropp_Ropp (IZR (Int_part r2))) in H5;
Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1)))
(Rplus (Ropp r2) (IZR (Int_part r2)))) in H5;
Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2)
(IZR (Int_part r2))) in H5;
Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (Ropp r2)) in H5;
Rewrite (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1)))
(IZR (Int_part r2))) in H5;
Rewrite <-(Rplus_assoc r1 (Ropp r2)
(Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))) in H5;
Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) in H5;
Fold (Rminus r1 r2) in H5;Fold (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))
in H5;Generalize (Rlt_compatibility
(Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) (Ropp R1)
(Rplus (Rminus r1 r2) (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) H5);
Intro;Clear H5;Rewrite (Rplus_sym (Rminus r1 r2)
(Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) in H;
Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) (Rminus r1 r2)) in H;
Unfold Rminus in H;Fold (Rminus r1 r2) in H;
Rewrite (Rplus_assoc (IZR (Int_part r1)) (Ropp (IZR (Int_part r2)))
(Rplus (IZR (Int_part r2)) (Ropp (IZR (Int_part r1))))) in H;
Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r2))) (IZR (Int_part r2))
(Ropp (IZR (Int_part r1)))) in H;Rewrite (Rplus_Ropp_l (IZR (Int_part r2))) in
H;Elim (Rplus_ne (Ropp (IZR (Int_part r1))));Intros a b;Rewrite b in H;
Clear a b;Rewrite (Rplus_Ropp_r (IZR (Int_part r1))) in H;
Elim (Rplus_ne (Rminus r1 r2));Intros a b;Rewrite b in H;
Clear a b;Fold (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) in H;
Fold (Rminus (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) R1) in H;
Rewrite (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))) in H1;
Rewrite (Ropp_Ropp (IZR (Int_part r2))) in H1;
Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1)))
(Rplus (Ropp r2) (IZR (Int_part r2)))) in H1;
Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2)
(IZR (Int_part r2))) in H1;
Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (Ropp r2)) in H1;
Rewrite (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1)))
(IZR (Int_part r2))) in H1;
Rewrite <-(Rplus_assoc r1 (Ropp r2)
(Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))) in H1;
Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) in H1;
Fold (Rminus r1 r2) in H1;Fold (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))
in H1;Generalize (Rlt_compatibility
(Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Rplus (Rminus r1 r2) (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) R0 H1);
Intro;Clear H1;
Rewrite (Rplus_sym (Rminus r1 r2)
(Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) in H0;
Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) (Rminus r1 r2)) in H0;
Rewrite <-(Ropp_distr2 (IZR (Int_part r1)) (IZR (Int_part r2))) in H0;
Rewrite (Rplus_Ropp_r (Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H0;
Elim (Rplus_ne (Rminus r1 r2));Intros a b;Rewrite b in H0;Clear a b;
Rewrite <-(Rplus_Ropp_l R1) in H0;
Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Ropp R1) R1) in H0;
Fold (Rminus (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) R1) in H0;
Rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0;
Rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H;
Cut R1==(IZR `1`);Auto with zarith real.
Intro;Rewrite H1 in H;Rewrite H1 in H0;Clear H1;
Rewrite (Z_R_minus `(Int_part r1)-(Int_part r2)` `1`) in H;
Rewrite (Z_R_minus `(Int_part r1)-(Int_part r2)` `1`) in H0;
Rewrite <-(plus_IZR `(Int_part r1)-(Int_part r2)-1` `1`) in H0;
Generalize (Rlt_le (IZR `(Int_part r1)-(Int_part r2)-1`) (Rminus r1 r2) H);
Intro;Clear H;
Generalize (up_tech (Rminus r1 r2) `(Int_part r1)-(Int_part r2)-1`
H1 H0);Intros;Clear H0 H1;Unfold 1 Int_part;Omega.
Qed.
(**********)
Lemma Rminus_fp1:(r1,r2:R)(Rge (frac_part r1) (frac_part r2))->
(frac_part (Rminus r1 r2))==(Rminus (frac_part r1) (frac_part r2)).
Intros;Unfold frac_part;
Generalize (Rminus_Int_part1 r1 r2 H);Intro;Rewrite -> H0;
Rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));Unfold Rminus;
Rewrite -> (Ropp_distr1 (IZR (Int_part r1)) (Ropp (IZR (Int_part r2))));
Rewrite -> (Ropp_distr1 r2 (Ropp (IZR (Int_part r2))));
Rewrite -> (Ropp_Ropp (IZR (Int_part r2)));
Rewrite -> (Rplus_assoc r1 (Ropp r2)
(Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))));
Rewrite -> (Rplus_assoc r1 (Ropp (IZR (Int_part r1)))
(Rplus (Ropp r2) (IZR (Int_part r2))));
Rewrite <- (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1)))
(IZR (Int_part r2)));
Rewrite <- (Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2)
(IZR (Int_part r2)));
Rewrite -> (Rplus_sym (Ropp r2) (Ropp (IZR (Int_part r1))));Auto with zarith real.
Qed.
(**********)
Lemma Rminus_fp2:(r1,r2:R)(Rlt (frac_part r1) (frac_part r2))->
(frac_part (Rminus r1 r2))==
(Rplus (Rminus (frac_part r1) (frac_part r2)) R1).
Intros;Unfold frac_part;Generalize (Rminus_Int_part2 r1 r2 H);Intro;
Rewrite -> H0;
Rewrite <- (Z_R_minus (Zminus (Int_part r1) (Int_part r2)) `1`);
Rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));Unfold Rminus;
Rewrite -> (Ropp_distr1 (Rplus (IZR (Int_part r1)) (Ropp (IZR (Int_part r2))))
(Ropp (IZR `1`)));
Rewrite -> (Ropp_distr1 r2 (Ropp (IZR (Int_part r2))));
Rewrite -> (Ropp_Ropp (IZR `1`));
Rewrite -> (Ropp_Ropp (IZR (Int_part r2)));
Rewrite -> (Ropp_distr1 (IZR (Int_part r1)));
Rewrite -> (Ropp_Ropp (IZR (Int_part r2)));Simpl;
Rewrite <- (Rplus_assoc (Rplus r1 (Ropp r2))
(Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) R1);
Rewrite -> (Rplus_assoc r1 (Ropp r2)
(Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))));
Rewrite -> (Rplus_assoc r1 (Ropp (IZR (Int_part r1)))
(Rplus (Ropp r2) (IZR (Int_part r2))));
Rewrite <- (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1)))
(IZR (Int_part r2)));
Rewrite <- (Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2)
(IZR (Int_part r2)));
Rewrite -> (Rplus_sym (Ropp r2) (Ropp (IZR (Int_part r1))));Auto with zarith real.
Qed.
(**********)
Lemma plus_Int_part1:(r1,r2:R)(Rge (Rplus (frac_part r1) (frac_part r2)) R1)->
(Int_part (Rplus r1 r2))=(Zplus (Zplus (Int_part r1) (Int_part r2)) `1`).
Intros;
Generalize (Rle_sym2 R1 (Rplus (frac_part r1) (frac_part r2)) H);
Intro;Clear H;Elim (base_fp r1);Elim (base_fp r2);Intros;Clear H H2;
Generalize (Rlt_compatibility (frac_part r2) (frac_part r1) R1 H3);
Intro;Clear H3;
Generalize (Rlt_compatibility R1 (frac_part r2) R1 H1);Intro;Clear H1;
Rewrite (Rplus_sym R1 (frac_part r2)) in H2;
Generalize (Rlt_trans (Rplus (frac_part r2) (frac_part r1))
(Rplus (frac_part r2) R1) (Rplus R1 R1) H H2);Intro;Clear H H2;
Rewrite (Rplus_sym (frac_part r2) (frac_part r1)) in H1;
Unfold frac_part in H0 H1;Unfold Rminus in H0 H1;
Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1)))
(Rplus r2 (Ropp (IZR (Int_part r2))))) in H1;
Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))) in H1;
Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))
r2) in H1;
Rewrite (Rplus_sym
(Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2) in H1;
Rewrite <-(Rplus_assoc r1 r2
(Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))) in H1;
Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))) in H1;
Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1)))
(Rplus r2 (Ropp (IZR (Int_part r2))))) in H0;
Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))) in H0;
Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))
r2) in H0;
Rewrite (Rplus_sym
(Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2) in H0;
Rewrite <-(Rplus_assoc r1 r2
(Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))) in H0;
Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))) in H0;
Generalize (Rle_compatibility (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))
R1 (Rplus (Rplus r1 r2)
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) H0);Intro;
Clear H0;
Generalize (Rlt_compatibility (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Rplus (Rplus r1 r2)
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) (Rplus R1 R1) H1);
Intro;Clear H1;
Rewrite (Rplus_sym (Rplus r1 r2)
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) in H;
Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) (Rplus r1 r2)) in H;
Rewrite (Rplus_Ropp_r (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H;
Elim (Rplus_ne (Rplus r1 r2));Intros a b;Rewrite b in H;Clear a b;
Rewrite (Rplus_sym (Rplus r1 r2)
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) in H0;
Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) (Rplus r1 r2)) in H0;
Rewrite (Rplus_Ropp_r (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H0;
Elim (Rplus_ne (Rplus r1 r2));Intros a b;Rewrite b in H0;Clear a b;
Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) R1 R1) in
H0;Cut R1==(IZR `1`);Auto with zarith real.
Intro;Rewrite H1 in H0;Rewrite H1 in H;Clear H1;
Rewrite <-(plus_IZR (Int_part r1) (Int_part r2)) in H;
Rewrite <-(plus_IZR (Int_part r1) (Int_part r2)) in H0;
Rewrite <-(plus_IZR `(Int_part r1)+(Int_part r2)` `1`) in H;
Rewrite <-(plus_IZR `(Int_part r1)+(Int_part r2)` `1`) in H0;
Rewrite <-(plus_IZR `(Int_part r1)+(Int_part r2)+1` `1`) in H0;
Generalize (up_tech (Rplus r1 r2) `(Int_part r1)+(Int_part r2)+1` H H0);Intro;
Clear H H0;Unfold 1 Int_part;Omega.
Qed.
(**********)
Lemma plus_Int_part2:(r1,r2:R)(Rlt (Rplus (frac_part r1) (frac_part r2)) R1)->
(Int_part (Rplus r1 r2))=(Zplus (Int_part r1) (Int_part r2)).
Intros;Elim (base_fp r1);Elim (base_fp r2);Intros;Clear H1 H3;
Generalize (Rle_sym2 R0 (frac_part r2) H0);Intro;Clear H0;
Generalize (Rle_sym2 R0 (frac_part r1) H2);Intro;Clear H2;
Generalize (Rle_compatibility (frac_part r1) R0 (frac_part r2) H1);
Intro;Clear H1;Elim (Rplus_ne (frac_part r1));Intros a b;
Rewrite a in H2;Clear a b;Generalize (Rle_trans R0 (frac_part r1)
(Rplus (frac_part r1) (frac_part r2)) H0 H2);Intro;Clear H0 H2;
Unfold frac_part in H H1;Unfold Rminus in H H1;
Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1)))
(Rplus r2 (Ropp (IZR (Int_part r2))))) in H1;
Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))) in H1;
Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))
r2) in H1;
Rewrite (Rplus_sym
(Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2) in H1;
Rewrite <-(Rplus_assoc r1 r2
(Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))) in H1;
Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))) in H1;
Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1)))
(Rplus r2 (Ropp (IZR (Int_part r2))))) in H;
Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))) in H;
Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))
r2) in H;
Rewrite (Rplus_sym
(Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2) in H;
Rewrite <-(Rplus_assoc r1 r2
(Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))) in H;
Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))) in H;
Generalize (Rle_compatibility (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))
R0 (Rplus (Rplus r1 r2)
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) H1);Intro;
Clear H1;
Generalize (Rlt_compatibility (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Rplus (Rplus r1 r2)
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) R1 H);
Intro;Clear H;
Rewrite (Rplus_sym (Rplus r1 r2)
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) in H1;
Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) (Rplus r1 r2)) in H1;
Rewrite (Rplus_Ropp_r (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H1;
Elim (Rplus_ne (Rplus r1 r2));Intros a b;Rewrite b in H1;Clear a b;
Rewrite (Rplus_sym (Rplus r1 r2)
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) in H0;
Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) (Rplus r1 r2)) in H0;
Rewrite (Rplus_Ropp_r (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H0;
Elim (Rplus_ne (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))));Intros a b;
Rewrite a in H0;Clear a b;Elim (Rplus_ne (Rplus r1 r2));Intros a b;
Rewrite b in H0;Clear a b;Cut R1==(IZR `1`);Auto with zarith real.
Intro;Rewrite H in H1;Clear H;
Rewrite <-(plus_IZR (Int_part r1) (Int_part r2)) in H0;
Rewrite <-(plus_IZR (Int_part r1) (Int_part r2)) in H1;
Rewrite <-(plus_IZR `(Int_part r1)+(Int_part r2)` `1`) in H1;
Generalize (up_tech (Rplus r1 r2) `(Int_part r1)+(Int_part r2)` H0 H1);Intro;
Clear H0 H1;Unfold 1 Int_part;Omega.
Qed.
(**********)
Lemma plus_frac_part1:(r1,r2:R)
(Rge (Rplus (frac_part r1) (frac_part r2)) R1)->
(frac_part (Rplus r1 r2))==
(Rminus (Rplus (frac_part r1) (frac_part r2)) R1).
Intros;Unfold frac_part;
Generalize (plus_Int_part1 r1 r2 H);Intro;Rewrite H0;
Rewrite (plus_IZR `(Int_part r1)+(Int_part r2)` `1`);
Rewrite (plus_IZR (Int_part r1) (Int_part r2));Simpl;Unfold 3 4 Rminus;
Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1)))
(Rplus r2 (Ropp (IZR (Int_part r2)))));
Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2))));
Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))
r2);
Rewrite (Rplus_sym
(Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2);
Rewrite <-(Rplus_assoc r1 r2
(Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))));
Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2)));
Unfold Rminus;
Rewrite (Rplus_assoc (Rplus r1 r2)
(Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))
(Ropp R1));
Rewrite <-(Ropp_distr1 (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) R1);
Trivial with zarith real.
Qed.
(**********)
Lemma plus_frac_part2:(r1,r2:R)
(Rlt (Rplus (frac_part r1) (frac_part r2)) R1)->
(frac_part (Rplus r1 r2))==(Rplus (frac_part r1) (frac_part r2)).
Intros;Unfold frac_part;
Generalize (plus_Int_part2 r1 r2 H);Intro;Rewrite H0;
Rewrite (plus_IZR (Int_part r1) (Int_part r2));Unfold 2 3 Rminus;
Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1)))
(Rplus r2 (Ropp (IZR (Int_part r2)))));
Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2))));
Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))
r2);
Rewrite (Rplus_sym
(Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2);
Rewrite <-(Rplus_assoc r1 r2
(Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))));
Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2)));Unfold Rminus;
Trivial with zarith real.
Qed.
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