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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: RIneq.v,v 1.2.2.1 2004/07/16 19:31:33 herbelin Exp $ i*)
-
-(***************************************************************************)
-(** Basic lemmas for the classical reals numbers *)
-(***************************************************************************)
-
-Require Export Raxioms.
-Require Export ZArithRing.
-Require Omega.
-Require Export Field.
-
-Open Local Scope Z_scope.
-Open Local Scope R_scope.
-
-Implicit Variable Type r:R.
-
-(***************************************************************************)
-(** Instantiating Ring tactic on reals *)
-(***************************************************************************)
-
-Lemma RTheory : (Ring_Theory Rplus Rmult R1 R0 Ropp [x,y:R]false).
- Split.
- Exact Rplus_sym.
- Symmetry; Apply Rplus_assoc.
- Exact Rmult_sym.
- Symmetry; Apply Rmult_assoc.
- Intro; Apply Rplus_Ol.
- Intro; Apply Rmult_1l.
- Exact Rplus_Ropp_r.
- Intros.
- Rewrite Rmult_sym.
- Rewrite (Rmult_sym n p).
- Rewrite (Rmult_sym m p).
- Apply Rmult_Rplus_distr.
- Intros; Contradiction.
-Defined.
-
-Add Field R Rplus Rmult R1 R0 Ropp [x,y:R]false Rinv RTheory Rinv_l
- with minus:=Rminus div:=Rdiv.
-
-(**************************************************************************)
-(** Relation between orders and equality *)
-(**************************************************************************)
-
-(**********)
-Lemma Rlt_antirefl:(r:R)~``r<r``.
- Generalize Rlt_antisym. Intuition EAuto.
-Qed.
-Hints Resolve Rlt_antirefl : real.
-
-Lemma Rle_refl : (x:R) ``x<=x``.
-Intro; Right; Reflexivity.
-Qed.
-
-Lemma Rlt_not_eq:(r1,r2:R)``r1<r2``->``r1<>r2``.
- Red; Intros r1 r2 H H0; Apply (Rlt_antirefl r1).
- Pattern 2 r1; Rewrite H0; Trivial.
-Qed.
-
-Lemma Rgt_not_eq:(r1,r2:R)``r1>r2``->``r1<>r2``.
-Intros; Apply sym_not_eqT; Apply Rlt_not_eq; Auto with real.
-Qed.
-
-(**********)
-Lemma imp_not_Req:(r1,r2:R)(``r1<r2``\/ ``r1>r2``) -> ``r1<>r2``.
-Generalize Rlt_not_eq Rgt_not_eq. Intuition EAuto.
-Qed.
-Hints Resolve imp_not_Req : real.
-
-(** Reasoning by case on equalities and order *)
-
-(**********)
-Lemma Req_EM:(r1,r2:R)(r1==r2)\/``r1<>r2``.
-Intros ; Generalize (total_order_T r1 r2) imp_not_Req ; Intuition EAuto 3.
-Qed.
-Hints Resolve Req_EM : real.
-
-(**********)
-Lemma total_order:(r1,r2:R)``r1<r2``\/(r1==r2)\/``r1>r2``.
-Intros;Generalize (total_order_T r1 r2);Tauto.
-Qed.
-
-(**********)
-Lemma not_Req:(r1,r2:R)``r1<>r2``->(``r1<r2``\/``r1>r2``).
-Intros; Generalize (total_order_T r1 r2) ; Tauto.
-Qed.
-
-
-(*********************************************************************************)
-(** Order Lemma : relating [<], [>], [<=] and [>=] *)
-(*********************************************************************************)
-
-(**********)
-Lemma Rlt_le:(r1,r2:R)``r1<r2``-> ``r1<=r2``.
-Intros ; Red ; Tauto.
-Qed.
-Hints Resolve Rlt_le : real.
-
-(**********)
-Lemma Rle_ge : (r1,r2:R)``r1<=r2`` -> ``r2>=r1``.
-NewDestruct 1; Red; Auto with real.
-Qed.
-
-Hints Immediate Rle_ge : real.
-
-(**********)
-Lemma Rge_le : (r1,r2:R)``r1>=r2`` -> ``r2<=r1``.
-NewDestruct 1; Red; Auto with real.
-Qed.
-
-Hints Resolve Rge_le : real.
-
-(**********)
-Lemma not_Rle:(r1,r2:R)~``r1<=r2`` -> ``r2<r1``.
-Intros r1 r2 ; Generalize (total_order r1 r2) ; Unfold Rle; Tauto.
-Qed.
-
-Hints Immediate not_Rle : real.
-
-Lemma not_Rge:(r1,r2:R)~``r1>=r2`` -> ``r1<r2``.
-Intros; Apply not_Rle; Auto with real.
-Qed.
-
-(**********)
-Lemma Rlt_le_not:(r1,r2:R)``r2<r1`` -> ~``r1<=r2``.
-Generalize Rlt_antisym imp_not_Req ; Unfold Rle.
-Intuition EAuto 3.
-Qed.
-
-Lemma Rle_not:(r1,r2:R)``r1>r2`` -> ~``r1<=r2``.
-Proof Rlt_le_not.
-
-Hints Immediate Rlt_le_not : real.
-
-Lemma Rle_not_lt: (r1, r2:R) ``r2 <= r1`` -> ~``r1<r2``.
-Intros r1 r2. Generalize (Rlt_antisym r1 r2) (imp_not_Req r1 r2).
-Unfold Rle; Intuition.
-Qed.
-
-(**********)
-Lemma Rlt_ge_not:(r1,r2:R)``r1<r2`` -> ~``r1>=r2``.
-Generalize Rlt_le_not. Unfold Rle Rge. Intuition EAuto 3.
-Qed.
-
-Hints Immediate Rlt_ge_not : real.
-
-(**********)
-Lemma eq_Rle:(r1,r2:R)r1==r2->``r1<=r2``.
-Unfold Rle; Tauto.
-Qed.
-Hints Immediate eq_Rle : real.
-
-Lemma eq_Rge:(r1,r2:R)r1==r2->``r1>=r2``.
-Unfold Rge; Tauto.
-Qed.
-Hints Immediate eq_Rge : real.
-
-Lemma eq_Rle_sym:(r1,r2:R)r2==r1->``r1<=r2``.
-Unfold Rle; Auto.
-Qed.
-Hints Immediate eq_Rle_sym : real.
-
-Lemma eq_Rge_sym:(r1,r2:R)r2==r1->``r1>=r2``.
-Unfold Rge; Auto.
-Qed.
-Hints Immediate eq_Rge_sym : real.
-
-Lemma Rle_antisym : (r1,r2:R)``r1<=r2`` -> ``r2<=r1``-> r1==r2.
-Intros r1 r2; Generalize (Rlt_antisym r1 r2) ; Unfold Rle ; Intuition.
-Qed.
-Hints Resolve Rle_antisym : real.
-
-(**********)
-Lemma Rle_le_eq:(r1,r2:R)(``r1<=r2``/\``r2<=r1``)<->(r1==r2).
-Intuition.
-Qed.
-
-Lemma Rlt_rew : (x,x',y,y':R)``x==x'``->``x'<y'`` -> `` y' == y`` -> ``x < y``.
-Intros x x' y y'; Intros; Replace x with x'; Replace y with y'; Assumption.
-Qed.
-
-(**********)
-Lemma Rle_trans:(r1,r2,r3:R) ``r1<=r2``->``r2<=r3``->``r1<=r3``.
-Generalize trans_eqT Rlt_trans Rlt_rew.
-Unfold Rle.
-Intuition EAuto 2.
-Qed.
-
-(**********)
-Lemma Rle_lt_trans:(r1,r2,r3:R)``r1<=r2``->``r2<r3``->``r1<r3``.
-Generalize Rlt_trans Rlt_rew.
-Unfold Rle.
-Intuition EAuto 2.
-Qed.
-
-(**********)
-Lemma Rlt_le_trans:(r1,r2,r3:R)``r1<r2``->``r2<=r3``->``r1<r3``.
-Generalize Rlt_trans Rlt_rew; Unfold Rle; Intuition EAuto 2.
-Qed.
-
-
-(** Decidability of the order *)
-Lemma total_order_Rlt:(r1,r2:R)(sumboolT ``r1<r2`` ~(``r1<r2``)).
-Intros;Generalize (total_order_T r1 r2) (imp_not_Req r1 r2) ; Intuition.
-Qed.
-
-(**********)
-Lemma total_order_Rle:(r1,r2:R)(sumboolT ``r1<=r2`` ~(``r1<=r2``)).
-Intros r1 r2.
-Generalize (total_order_T r1 r2) (imp_not_Req r1 r2).
-Intuition EAuto 4 with real.
-Qed.
-
-(**********)
-Lemma total_order_Rgt:(r1,r2:R)(sumboolT ``r1>r2`` ~(``r1>r2``)).
-Intros;Unfold Rgt;Intros;Apply total_order_Rlt.
-Qed.
-
-(**********)
-Lemma total_order_Rge:(r1,r2:R)(sumboolT (``r1>=r2``) ~(``r1>=r2``)).
-Intros;Generalize (total_order_Rle r2 r1);Intuition.
-Qed.
-
-Lemma total_order_Rlt_Rle:(r1,r2:R)(sumboolT ``r1<r2`` ``r2<=r1``).
-Intros;Generalize (total_order_T r1 r2); Intuition.
-Qed.
-
-Lemma Rle_or_lt: (n, m:R)(Rle n m) \/ (Rlt m n).
-Intros n m; Elim (total_order_Rlt_Rle m n);Auto with real.
-Qed.
-
-Lemma total_order_Rle_Rlt_eq :(r1,r2:R)``r1<=r2``->
- (sumboolT ``r1<r2`` ``r1==r2``).
-Intros r1 r2 H;Generalize (total_order_T r1 r2); Intuition.
-Qed.
-
-(**********)
-Lemma inser_trans_R:(n,m,p,q:R)``n<=m<p``-> (sumboolT ``n<=m<q`` ``q<=m<p``).
-Intros n m p q; Intros; Generalize (total_order_Rlt_Rle m q); Intuition.
-Qed.
-
-(****************************************************************)
-(** Field Lemmas *)
-(* This part contains lemma involving the Fields operations *)
-(****************************************************************)
-(*********************************************************)
-(** Addition *)
-(*********************************************************)
-
-Lemma Rplus_ne:(r:R)``r+0==r``/\``0+r==r``.
-Intro;Split;Ring.
-Qed.
-Hints Resolve Rplus_ne : real v62.
-
-Lemma Rplus_Or:(r:R)``r+0==r``.
-Intro; Ring.
-Qed.
-Hints Resolve Rplus_Or : real.
-
-(**********)
-Lemma Rplus_Ropp_l:(r:R)``(-r)+r==0``.
- Intro; Ring.
-Qed.
-Hints Resolve Rplus_Ropp_l : real.
-
-
-(**********)
-Lemma Rplus_Ropp:(x,y:R)``x+y==0``->``y== -x``.
- Intros x y H; Replace y with ``(-x+x)+y``;
- [ Rewrite -> Rplus_assoc; Rewrite -> H; Ring
- | Ring ].
-Qed.
-
-(*i New i*)
-Hint eqT_R_congr : real := Resolve (congr_eqT R).
-
-Lemma Rplus_plus_r:(r,r1,r2:R)(r1==r2)->``r+r1==r+r2``.
- Auto with real.
-Qed.
-
-(*i Old i*)Hints Resolve Rplus_plus_r : v62.
-
-(**********)
-Lemma r_Rplus_plus:(r,r1,r2:R)``r+r1==r+r2``->r1==r2.
- Intros; Transitivity ``(-r+r)+r1``.
- Ring.
- Transitivity ``(-r+r)+r2``.
- Repeat Rewrite -> Rplus_assoc; Rewrite <- H; Reflexivity.
- Ring.
-Qed.
-Hints Resolve r_Rplus_plus : real.
-
-(**********)
-Lemma Rplus_ne_i:(r,b:R)``r+b==r`` -> ``b==0``.
- Intros r b; Pattern 2 r; Replace r with ``r+0``;
- EAuto with real.
-Qed.
-
-(***********************************************************)
-(** Multiplication *)
-(***********************************************************)
-
-(**********)
-Lemma Rinv_r:(r:R)``r<>0``->``r* (/r)==1``.
- Intros; Rewrite -> Rmult_sym; Auto with real.
-Qed.
-Hints Resolve Rinv_r : real.
-
-Lemma Rinv_l_sym:(r:R)``r<>0``->``1==(/r) * r``.
- Symmetry; Auto with real.
-Qed.
-
-Lemma Rinv_r_sym:(r:R)``r<>0``->``1==r* (/r)``.
- Symmetry; Auto with real.
-Qed.
-Hints Resolve Rinv_l_sym Rinv_r_sym : real.
-
-
-(**********)
-Lemma Rmult_Or :(r:R) ``r*0==0``.
-Intro; Ring.
-Qed.
-Hints Resolve Rmult_Or : real v62.
-
-(**********)
-Lemma Rmult_Ol:(r:R) ``0*r==0``.
-Intro; Ring.
-Qed.
-Hints Resolve Rmult_Ol : real v62.
-
-(**********)
-Lemma Rmult_ne:(r:R)``r*1==r``/\``1*r==r``.
-Intro;Split;Ring.
-Qed.
-Hints Resolve Rmult_ne : real v62.
-
-(**********)
-Lemma Rmult_1r:(r:R)(``r*1==r``).
-Intro; Ring.
-Qed.
-Hints Resolve Rmult_1r : real.
-
-(**********)
-Lemma Rmult_mult_r:(r,r1,r2:R)r1==r2->``r*r1==r*r2``.
- Auto with real.
-Qed.
-
-(*i OLD i*)Hints Resolve Rmult_mult_r : v62.
-
-(**********)
-Lemma r_Rmult_mult:(r,r1,r2:R)(``r*r1==r*r2``)->``r<>0``->(r1==r2).
- Intros; Transitivity ``(/r * r)*r1``.
- Rewrite Rinv_l; Auto with real.
- Transitivity ``(/r * r)*r2``.
- Repeat Rewrite Rmult_assoc; Rewrite H; Trivial.
- Rewrite Rinv_l; Auto with real.
-Qed.
-
-(**********)
-Lemma without_div_Od:(r1,r2:R)``r1*r2==0`` -> ``r1==0`` \/ ``r2==0``.
- Intros; Case (Req_EM r1 ``0``); [Intro Hz | Intro Hnotz].
- Auto.
- Right; Apply r_Rmult_mult with r1; Trivial.
- Rewrite H; Auto with real.
-Qed.
-
-(**********)
-Lemma without_div_Oi:(r1,r2:R) ``r1==0``\/``r2==0`` -> ``r1*r2==0``.
- Intros r1 r2 [H | H]; Rewrite H; Auto with real.
-Qed.
-
-Hints Resolve without_div_Oi : real.
-
-(**********)
-Lemma without_div_Oi1:(r1,r2:R) ``r1==0`` -> ``r1*r2==0``.
- Auto with real.
-Qed.
-
-(**********)
-Lemma without_div_Oi2:(r1,r2:R) ``r2==0`` -> ``r1*r2==0``.
- Auto with real.
-Qed.
-
-
-(**********)
-Lemma without_div_O_contr:(r1,r2:R)``r1*r2<>0`` -> ``r1<>0`` /\ ``r2<>0``.
-Intros r1 r2 H; Split; Red; Intro; Apply H; Auto with real.
-Qed.
-
-(**********)
-Lemma mult_non_zero :(r1,r2:R)``r1<>0`` /\ ``r2<>0`` -> ``r1*r2<>0``.
-Red; Intros r1 r2 (H1,H2) H.
-Case (without_div_Od r1 r2); Auto with real.
-Qed.
-Hints Resolve mult_non_zero : real.
-
-(**********)
-Lemma Rmult_Rplus_distrl:
- (r1,r2,r3:R) ``(r1+r2)*r3 == (r1*r3)+(r2*r3)``.
-Intros; Ring.
-Qed.
-
-(** Square function *)
-
-(***********)
-Definition Rsqr:R->R:=[r:R]``r*r``.
-V7only[Notation "x ²" := (Rsqr x) (at level 2,left associativity).].
-
-(***********)
-Lemma Rsqr_O:(Rsqr ``0``)==``0``.
- Unfold Rsqr; Auto with real.
-Qed.
-
-(***********)
-Lemma Rsqr_r_R0:(r:R)(Rsqr r)==``0``->``r==0``.
-Unfold Rsqr;Intros;Elim (without_div_Od r r H);Trivial.
-Qed.
-
-(*********************************************************)
-(** Opposite *)
-(*********************************************************)
-
-(**********)
-Lemma eq_Ropp:(r1,r2:R)(r1==r2)->``-r1 == -r2``.
- Auto with real.
-Qed.
-Hints Resolve eq_Ropp : real.
-
-(**********)
-Lemma Ropp_O:``-0==0``.
- Ring.
-Qed.
-Hints Resolve Ropp_O : real v62.
-
-(**********)
-Lemma eq_RoppO:(r:R)``r==0``-> ``-r==0``.
- Intros; Rewrite -> H; Auto with real.
-Qed.
-Hints Resolve eq_RoppO : real.
-
-(**********)
-Lemma Ropp_Ropp:(r:R)``-(-r)==r``.
- Intro; Ring.
-Qed.
-Hints Resolve Ropp_Ropp : real.
-
-(*********)
-Lemma Ropp_neq:(r:R)``r<>0``->``-r<>0``.
-Red;Intros r H H0.
-Apply H.
-Transitivity ``-(-r)``; Auto with real.
-Qed.
-Hints Resolve Ropp_neq : real.
-
-(**********)
-Lemma Ropp_distr1:(r1,r2:R)``-(r1+r2)==(-r1 + -r2)``.
- Intros; Ring.
-Qed.
-Hints Resolve Ropp_distr1 : real.
-
-(** Opposite and multiplication *)
-
-Lemma Ropp_mul1:(r1,r2:R)``(-r1)*r2 == -(r1*r2)``.
- Intros; Ring.
-Qed.
-Hints Resolve Ropp_mul1 : real.
-
-(**********)
-Lemma Ropp_mul2:(r1,r2:R)``(-r1)*(-r2)==r1*r2``.
- Intros; Ring.
-Qed.
-Hints Resolve Ropp_mul2 : real.
-
-Lemma Ropp_mul3 : (r1,r2:R) ``r1*(-r2) == -(r1*r2)``.
-Intros; Rewrite <- Ropp_mul1; Ring.
-Qed.
-
-(** Substraction *)
-
-Lemma minus_R0:(r:R)``r-0==r``.
-Intro;Ring.
-Qed.
-Hints Resolve minus_R0 : real.
-
-Lemma Rminus_Ropp:(r:R)``0-r==-r``.
-Intro;Ring.
-Qed.
-Hints Resolve Rminus_Ropp : real.
-
-(**********)
-Lemma Ropp_distr2:(r1,r2:R)``-(r1-r2)==r2-r1``.
- Intros; Ring.
-Qed.
-Hints Resolve Ropp_distr2 : real.
-
-Lemma Ropp_distr3:(r1,r2:R)``-(r2-r1)==r1-r2``.
-Intros; Ring.
-Qed.
-Hints Resolve Ropp_distr3 : real.
-
-(**********)
-Lemma eq_Rminus:(r1,r2:R)(r1==r2)->``r1-r2==0``.
- Intros; Rewrite H; Ring.
-Qed.
-Hints Resolve eq_Rminus : real.
-
-(**********)
-Lemma Rminus_eq:(r1,r2:R)``r1-r2==0`` -> r1==r2.
- Intros r1 r2; Unfold Rminus; Rewrite -> Rplus_sym; Intro.
- Rewrite <- (Ropp_Ropp r2); Apply (Rplus_Ropp (Ropp r2) r1 H).
-Qed.
-Hints Immediate Rminus_eq : real.
-
-Lemma Rminus_eq_right:(r1,r2:R)``r2-r1==0`` -> r1==r2.
-Intros;Generalize (Rminus_eq r2 r1 H);Clear H;Intro H;Rewrite H;Ring.
-Qed.
-Hints Immediate Rminus_eq_right : real.
-
-Lemma Rplus_Rminus: (p,q:R)``p+(q-p)``==q.
-Intros; Ring.
-Qed.
-Hints Resolve Rplus_Rminus:real.
-
-(**********)
-Lemma Rminus_eq_contra:(r1,r2:R)``r1<>r2``->``r1-r2<>0``.
-Red; Intros r1 r2 H H0.
-Apply H; Auto with real.
-Qed.
-Hints Resolve Rminus_eq_contra : real.
-
-Lemma Rminus_not_eq:(r1,r2:R)``r1-r2<>0``->``r1<>r2``.
-Red; Intros; Elim H; Apply eq_Rminus; Auto.
-Qed.
-Hints Resolve Rminus_not_eq : real.
-
-Lemma Rminus_not_eq_right:(r1,r2:R)``r2-r1<>0`` -> ``r1<>r2``.
-Red; Intros;Elim H;Rewrite H0; Ring.
-Qed.
-Hints Resolve Rminus_not_eq_right : real.
-
-V7only [Notation not_sym := (sym_not_eq R).].
-
-(**********)
-Lemma Rminus_distr: (x,y,z:R) ``x*(y-z)==(x*y) - (x*z)``.
-Intros; Ring.
-Qed.
-
-(** Inverse *)
-Lemma Rinv_R1:``/1==1``.
-Field;Auto with real.
-Qed.
-Hints Resolve Rinv_R1 : real.
-
-(*********)
-Lemma Rinv_neq_R0:(r:R)``r<>0``->``(/r)<>0``.
-Red; Intros; Apply R1_neq_R0.
-Replace ``1`` with ``(/r) * r``; Auto with real.
-Qed.
-Hints Resolve Rinv_neq_R0 : real.
-
-(*********)
-Lemma Rinv_Rinv:(r:R)``r<>0``->``/(/r)==r``.
-Intros;Field;Auto with real.
-Qed.
-Hints Resolve Rinv_Rinv : real.
-
-(*********)
-Lemma Rinv_Rmult:(r1,r2:R)``r1<>0``->``r2<>0``->``/(r1*r2)==(/r1)*(/r2)``.
-Intros;Field;Auto with real.
-Qed.
-
-(*********)
-Lemma Ropp_Rinv:(r:R)``r<>0``->``-(/r)==/(-r)``.
-Intros;Field;Auto with real.
-Qed.
-
-Lemma Rinv_r_simpl_r : (r1,r2:R)``r1<>0``->``r1*(/r1)*r2==r2``.
-Intros; Transitivity ``1*r2``; Auto with real.
-Rewrite Rinv_r; Auto with real.
-Qed.
-
-Lemma Rinv_r_simpl_l : (r1,r2:R)``r1<>0``->``r2*r1*(/r1)==r2``.
-Intros; Transitivity ``r2*1``; Auto with real.
-Transitivity ``r2*(r1*/r1)``; Auto with real.
-Qed.
-
-Lemma Rinv_r_simpl_m : (r1,r2:R)``r1<>0``->``r1*r2*(/r1)==r2``.
-Intros; Transitivity ``r2*1``; Auto with real.
-Transitivity ``r2*(r1*/r1)``; Auto with real.
-Ring.
-Qed.
-Hints Resolve Rinv_r_simpl_l Rinv_r_simpl_r Rinv_r_simpl_m : real.
-
-(*********)
-Lemma Rinv_Rmult_simpl:(a,b,c:R)``a<>0``->``(a*(/b))*(c*(/a))==c*(/b)``.
-Intros a b c; Intros.
-Transitivity ``(a*/a)*(c*(/b))``; Auto with real.
-Ring.
-Qed.
-
-(** Order and addition *)
-
-Lemma Rlt_compatibility_r:(r,r1,r2:R)``r1<r2``->``r1+r<r2+r``.
-Intros.
-Rewrite (Rplus_sym r1 r); Rewrite (Rplus_sym r2 r); Auto with real.
-Qed.
-
-Hints Resolve Rlt_compatibility_r : real.
-
-(**********)
-Lemma Rlt_anti_compatibility: (r,r1,r2:R)``r+r1 < r+r2`` -> ``r1<r2``.
-Intros; Cut ``(-r+r)+r1 < (-r+r)+r2``.
-Rewrite -> Rplus_Ropp_l.
-Elim (Rplus_ne r1); Elim (Rplus_ne r2); Intros; Rewrite <- H3;
- Rewrite <- H1; Auto with zarith real.
-Rewrite -> Rplus_assoc; Rewrite -> Rplus_assoc;
- Apply (Rlt_compatibility ``-r`` ``r+r1`` ``r+r2`` H).
-Qed.
-
-(**********)
-Lemma Rle_compatibility:(r,r1,r2:R)``r1<=r2`` -> ``r+r1 <= r+r2 ``.
-Unfold Rle; Intros; Elim H; Intro.
-Left; Apply (Rlt_compatibility r r1 r2 H0).
-Right; Rewrite <- H0; Auto with zarith real.
-Qed.
-
-(**********)
-Lemma Rle_compatibility_r:(r,r1,r2:R)``r1<=r2`` -> ``r1+r<=r2+r``.
-Unfold Rle; Intros; Elim H; Intro.
-Left; Apply (Rlt_compatibility_r r r1 r2 H0).
-Right; Rewrite <- H0; Auto with real.
-Qed.
-
-Hints Resolve Rle_compatibility Rle_compatibility_r : real.
-
-(**********)
-Lemma Rle_anti_compatibility: (r,r1,r2:R)``r+r1<=r+r2`` -> ``r1<=r2``.
-Unfold Rle; Intros; Elim H; Intro.
-Left; Apply (Rlt_anti_compatibility r r1 r2 H0).
-Right; Apply (r_Rplus_plus r r1 r2 H0).
-Qed.
-
-(**********)
-Lemma sum_inequa_Rle_lt:(a,x,b,c,y,d:R)``a<=x`` -> ``x<b`` ->
- ``c<y`` -> ``y<=d`` -> ``a+c < x+y < b+d``.
-Intros;Split.
-Apply Rlt_le_trans with ``a+y``; Auto with real.
-Apply Rlt_le_trans with ``b+y``; Auto with real.
-Qed.
-
-(*********)
-Lemma Rplus_lt:(r1,r2,r3,r4:R)``r1<r2`` -> ``r3<r4`` -> ``r1+r3 < r2+r4``.
-Intros; Apply Rlt_trans with ``r2+r3``; Auto with real.
-Qed.
-
-Lemma Rplus_le:(r1,r2,r3,r4:R)``r1<=r2`` -> ``r3<=r4`` -> ``r1+r3 <= r2+r4``.
-Intros; Apply Rle_trans with ``r2+r3``; Auto with real.
-Qed.
-
-(*********)
-Lemma Rplus_lt_le_lt:(r1,r2,r3,r4:R)``r1<r2`` -> ``r3<=r4`` ->
- ``r1+r3 < r2+r4``.
-Intros; Apply Rlt_le_trans with ``r2+r3``; Auto with real.
-Qed.
-
-(*********)
-Lemma Rplus_le_lt_lt:(r1,r2,r3,r4:R)``r1<=r2`` -> ``r3<r4`` ->
- ``r1+r3 < r2+r4``.
-Intros; Apply Rle_lt_trans with ``r2+r3``; Auto with real.
-Qed.
-
-Hints Immediate Rplus_lt Rplus_le Rplus_lt_le_lt Rplus_le_lt_lt : real.
-
-(** Order and Opposite *)
-
-(**********)
-Lemma Rgt_Ropp:(r1,r2:R) ``r1 > r2`` -> ``-r1 < -r2``.
-Unfold Rgt; Intros.
-Apply (Rlt_anti_compatibility ``r2+r1``).
-Replace ``r2+r1+(-r1)`` with r2.
-Replace ``r2+r1+(-r2)`` with r1.
-Trivial.
-Ring.
-Ring.
-Qed.
-Hints Resolve Rgt_Ropp.
-
-(**********)
-Lemma Rlt_Ropp:(r1,r2:R) ``r1 < r2`` -> ``-r1 > -r2``.
-Unfold Rgt; Auto with real.
-Qed.
-Hints Resolve Rlt_Ropp : real.
-
-Lemma Ropp_Rlt: (x,y:R) ``-y < -x`` ->``x<y``.
-Intros x y H'.
-Rewrite <- (Ropp_Ropp x); Rewrite <- (Ropp_Ropp y); Auto with real.
-Qed.
-Hints Immediate Ropp_Rlt : real.
-
-Lemma Rlt_Ropp1:(r1,r2:R) ``r2 < r1`` -> ``-r1 < -r2``.
-Auto with real.
-Qed.
-Hints Resolve Rlt_Ropp1 : real.
-
-(**********)
-Lemma Rle_Ropp:(r1,r2:R) ``r1 <= r2`` -> ``-r1 >= -r2``.
-Unfold Rge; Intros r1 r2 [H|H]; Auto with real.
-Qed.
-Hints Resolve Rle_Ropp : real.
-
-Lemma Ropp_Rle: (x,y:R) ``-y <= -x`` ->``x <= y``.
-Intros x y H.
-Elim H;Auto with real.
-Intro H1;Rewrite <-(Ropp_Ropp x);Rewrite <-(Ropp_Ropp y);Rewrite H1;
- Auto with real.
-Qed.
-Hints Immediate Ropp_Rle : real.
-
-Lemma Rle_Ropp1:(r1,r2:R) ``r2 <= r1`` -> ``-r1 <= -r2``.
-Intros r1 r2 H;Elim H;Auto with real.
-Qed.
-Hints Resolve Rle_Ropp1 : real.
-
-(**********)
-Lemma Rge_Ropp:(r1,r2:R) ``r1 >= r2`` -> ``-r1 <= -r2``.
-Unfold Rge; Intros r1 r2 [H|H]; Auto with real.
-Qed.
-Hints Resolve Rge_Ropp : real.
-
-(**********)
-Lemma Rlt_RO_Ropp:(r:R) ``0 < r`` -> ``0 > -r``.
-Intros; Replace ``0`` with ``-0``; Auto with real.
-Qed.
-Hints Resolve Rlt_RO_Ropp : real.
-
-(**********)
-Lemma Rgt_RO_Ropp:(r:R) ``0 > r`` -> ``0 < -r``.
-Intros; Replace ``0`` with ``-0``; Auto with real.
-Qed.
-Hints Resolve Rgt_RO_Ropp : real.
-
-(**********)
-Lemma Rgt_RoppO:(r:R)``r>0``->``(-r)<0``.
-Intros; Rewrite <- Ropp_O; Auto with real.
-Qed.
-
-(**********)
-Lemma Rlt_RoppO:(r:R)``r<0``->``-r>0``.
-Intros; Rewrite <- Ropp_O; Auto with real.
-Qed.
-Hints Resolve Rgt_RoppO Rlt_RoppO: real.
-
-(**********)
-Lemma Rle_RO_Ropp:(r:R) ``0 <= r`` -> ``0 >= -r``.
-Intros; Replace ``0`` with ``-0``; Auto with real.
-Qed.
-Hints Resolve Rle_RO_Ropp : real.
-
-(**********)
-Lemma Rge_RO_Ropp:(r:R) ``0 >= r`` -> ``0 <= -r``.
-Intros; Replace ``0`` with ``-0``; Auto with real.
-Qed.
-Hints Resolve Rge_RO_Ropp : real.
-
-(** Order and multiplication *)
-
-Lemma Rlt_monotony_r:(r,r1,r2:R)``0<r`` -> ``r1 < r2`` -> ``r1*r < r2*r``.
-Intros; Rewrite (Rmult_sym r1 r); Rewrite (Rmult_sym r2 r); Auto with real.
-Qed.
-Hints Resolve Rlt_monotony_r.
-
-Lemma Rlt_monotony_contra: (z, x, y:R) ``0<z`` ->``z*x<z*y`` ->``x<y``.
-Intros z x y H H0.
-Case (total_order x y); Intros Eq0; Auto; Elim Eq0; Clear Eq0; Intros Eq0.
- Rewrite Eq0 in H0;ElimType False;Apply (Rlt_antirefl ``z*y``);Auto.
-Generalize (Rlt_monotony z y x H Eq0);Intro;ElimType False;
- Generalize (Rlt_trans ``z*x`` ``z*y`` ``z*x`` H0 H1);Intro;
- Apply (Rlt_antirefl ``z*x``);Auto.
-Qed.
-
-V7only [
-Notation Rlt_monotony_rev := Rlt_monotony_contra.
-Notation "'Rlt_monotony_contra' a b c" := (Rlt_monotony_contra c a b)
- (at level 10, a,b,c at level 9, only parsing).
-].
-
-Lemma Rlt_anti_monotony:(r,r1,r2:R)``r < 0`` -> ``r1 < r2`` -> ``r*r1 > r*r2``.
-Intros; Replace r with ``-(-r)``; Auto with real.
-Rewrite (Ropp_mul1 ``-r``); Rewrite (Ropp_mul1 ``-r``).
-Apply Rlt_Ropp; Auto with real.
-Qed.
-
-(**********)
-Lemma Rle_monotony:
- (r,r1,r2:R)``0 <= r`` -> ``r1 <= r2`` -> ``r*r1 <= r*r2``.
-Intros r r1 r2 H H0; NewDestruct H; NewDestruct H0; Unfold Rle; Auto with real.
-Right; Rewrite <- H; Do 2 Rewrite Rmult_Ol; Reflexivity.
-Qed.
-Hints Resolve Rle_monotony : real.
-
-Lemma Rle_monotony_r:
- (r,r1,r2:R)``0 <= r`` -> ``r1 <= r2`` -> ``r1*r <= r2*r``.
-Intros r r1 r2 H;
-Rewrite (Rmult_sym r1 r); Rewrite (Rmult_sym r2 r); Auto with real.
-Qed.
-Hints Resolve Rle_monotony_r : real.
-
-Lemma Rmult_le_reg_l:
- (z, x, y:R) ``0<z`` ->``z*x<=z*y`` ->``x<=y``.
-Intros z x y H H0;Case H0; Auto with real.
-Intros H1; Apply Rlt_le.
-Apply Rlt_monotony_contra with z := z;Auto.
-Intros H1;Replace x with (Rmult (Rinv z) (Rmult z x)); Auto with real.
-Replace y with (Rmult (Rinv z) (Rmult z y)).
- Rewrite H1;Auto with real.
-Rewrite <- Rmult_assoc; Rewrite Rinv_l; Auto with real.
-Rewrite <- Rmult_assoc; Rewrite Rinv_l; Auto with real.
-Qed.
-
-V7only [
-Notation "'Rle_monotony_contra' a b c" := (Rmult_le_reg_l c a b)
- (at level 10, a,b,c at level 9, only parsing).
-Notation Rle_monotony_contra := Rmult_le_reg_l.
-].
-
-
-Lemma Rle_anti_monotony1
- :(r,r1,r2:R)``r <= 0`` -> ``r1 <= r2`` -> ``r*r2 <= r*r1``.
-Intros; Replace r with ``-(-r)``; Auto with real.
-Do 2 Rewrite (Ropp_mul1 ``-r``).
-Apply Rle_Ropp1; Auto with real.
-Qed.
-Hints Resolve Rle_anti_monotony1 : real.
-
-Lemma Rle_anti_monotony
- :(r,r1,r2:R)``r <= 0`` -> ``r1 <= r2`` -> ``r*r1 >= r*r2``.
-Intros; Apply Rle_ge; Auto with real.
-Qed.
-Hints Resolve Rle_anti_monotony : real.
-
-Lemma Rle_Rmult_comp:
- (x, y, z, t:R) ``0 <= x`` -> ``0 <= z`` -> ``x <= y`` -> ``z <= t`` ->
- ``x*z <= y*t``.
-Intros x y z t H' H'0 H'1 H'2.
-Apply Rle_trans with r2 := ``x*t``; Auto with real.
-Repeat Rewrite [x:?](Rmult_sym x t).
-Apply Rle_monotony; Auto.
-Apply Rle_trans with z; Auto.
-Qed.
-Hints Resolve Rle_Rmult_comp :real.
-
-Lemma Rmult_lt:(r1,r2,r3,r4:R)``r3>0`` -> ``r2>0`` ->
- `` r1 < r2`` -> ``r3 < r4`` -> ``r1*r3 < r2*r4``.
-Intros; Apply Rlt_trans with ``r2*r3``; Auto with real.
-Qed.
-
-(*********)
-Lemma Rmult_lt_0
- :(r1,r2,r3,r4:R)``r3>=0``->``r2>0``->``r1<r2``->``r3<r4``->``r1*r3<r2*r4``.
-Intros; Apply Rle_lt_trans with ``r2*r3``; Auto with real.
-Qed.
-
-(** Order and Substractions *)
-Lemma Rlt_minus:(r1,r2:R)``r1 < r2`` -> ``r1-r2 < 0``.
-Intros; Apply (Rlt_anti_compatibility ``r2``).
-Replace ``r2+(r1-r2)`` with r1.
-Replace ``r2+0`` with r2; Auto with real.
-Ring.
-Qed.
-Hints Resolve Rlt_minus : real.
-
-(**********)
-Lemma Rle_minus:(r1,r2:R)``r1 <= r2`` -> ``r1-r2 <= 0``.
-NewDestruct 1; Unfold Rle; Auto with real.
-Qed.
-
-(**********)
-Lemma Rminus_lt:(r1,r2:R)``r1-r2 < 0`` -> ``r1 < r2``.
-Intros; Replace r1 with ``r1-r2+r2``.
-Pattern 3 r2; Replace r2 with ``0+r2``; Auto with real.
-Ring.
-Qed.
-
-(**********)
-Lemma Rminus_le:(r1,r2:R)``r1-r2 <= 0`` -> ``r1 <= r2``.
-Intros; Replace r1 with ``r1-r2+r2``.
-Pattern 3 r2; Replace r2 with ``0+r2``; Auto with real.
-Ring.
-Qed.
-
-(**********)
-Lemma tech_Rplus:(r,s:R)``0<=r`` -> ``0<s`` -> ``r+s<>0``.
-Intros; Apply sym_not_eqT; Apply Rlt_not_eq.
-Rewrite Rplus_sym; Replace ``0`` with ``0+0``; Auto with real.
-Qed.
-Hints Immediate tech_Rplus : real.
-
-(** Order and the square function *)
-Lemma pos_Rsqr:(r:R)``0<=(Rsqr r)``.
-Intro; Case (total_order_Rlt_Rle r ``0``); Unfold Rsqr; Intro.
-Replace ``r*r`` with ``(-r)*(-r)``; Auto with real.
-Replace ``0`` with ``-r*0``; Auto with real.
-Replace ``0`` with ``0*r``; Auto with real.
-Qed.
-
-(***********)
-Lemma pos_Rsqr1:(r:R)``r<>0``->``0<(Rsqr r)``.
-Intros; Case (not_Req r ``0``); Trivial; Unfold Rsqr; Intro.
-Replace ``r*r`` with ``(-r)*(-r)``; Auto with real.
-Replace ``0`` with ``-r*0``; Auto with real.
-Replace ``0`` with ``0*r``; Auto with real.
-Qed.
-Hints Resolve pos_Rsqr pos_Rsqr1 : real.
-
-(** Zero is less than one *)
-Lemma Rlt_R0_R1:``0<1``.
-Replace ``1`` with ``(Rsqr 1)``; Auto with real.
-Unfold Rsqr; Auto with real.
-Qed.
-Hints Resolve Rlt_R0_R1 : real.
-
-Lemma Rle_R0_R1:``0<=1``.
-Left.
-Exact Rlt_R0_R1.
-Qed.
-
-(** Order and inverse *)
-Lemma Rlt_Rinv:(r:R)``0<r``->``0</r``.
-Intros; Apply not_Rle; Red; Intros.
-Absurd ``1<=0``; Auto with real.
-Replace ``1`` with ``r*(/r)``; Auto with real.
-Replace ``0`` with ``r*0``; Auto with real.
-Qed.
-Hints Resolve Rlt_Rinv : real.
-
-(*********)
-Lemma Rlt_Rinv2:(r:R)``r < 0``->``/r < 0``.
-Intros; Apply not_Rle; Red; Intros.
-Absurd ``1<=0``; Auto with real.
-Replace ``1`` with ``r*(/r)``; Auto with real.
-Replace ``0`` with ``r*0``; Auto with real.
-Qed.
-Hints Resolve Rlt_Rinv2 : real.
-
-(*********)
-Lemma Rinv_lt:(r1,r2:R)``0 < r1*r2`` -> ``r1 < r2`` -> ``/r2 < /r1``.
-Intros; Apply Rlt_monotony_rev with ``r1*r2``; Auto with real.
-Case (without_div_O_contr r1 r2 ); Intros; Auto with real.
-Replace ``r1*r2*/r2`` with r1.
-Replace ``r1*r2*/r1`` with r2; Trivial.
-Symmetry; Auto with real.
-Symmetry; Auto with real.
-Qed.
-
-Lemma Rlt_Rinv_R1: (x, y:R) ``1 <= x`` -> ``x<y`` ->``/y< /x``.
-Intros x y H' H'0.
-Cut (Rlt R0 x); [Intros Lt0 | Apply Rlt_le_trans with r2 := R1];
- Auto with real.
-Apply Rlt_monotony_contra with z := x; Auto with real.
-Rewrite (Rmult_sym x (Rinv x)); Rewrite Rinv_l; Auto with real.
-Apply Rlt_monotony_contra with z := y; Auto with real.
-Apply Rlt_trans with r2:=x;Auto.
-Cut ``y*(x*/y)==x``.
-Intro H1;Rewrite H1;Rewrite (Rmult_1r y);Auto.
-Rewrite (Rmult_sym x); Rewrite <- Rmult_assoc; Rewrite (Rmult_sym y (Rinv y));
- Rewrite Rinv_l; Auto with real.
-Apply imp_not_Req; Right.
-Red; Apply Rlt_trans with r2 := x; Auto with real.
-Qed.
-Hints Resolve Rlt_Rinv_R1 :real.
-
-(*********************************************************)
-(** Greater *)
-(*********************************************************)
-
-(**********)
-Lemma Rge_ge_eq:(r1,r2:R)``r1 >= r2`` -> ``r2 >= r1`` -> r1==r2.
-Intros; Apply Rle_antisym; Auto with real.
-Qed.
-
-(**********)
-Lemma Rlt_not_ge:(r1,r2:R)~(``r1<r2``)->``r1>=r2``.
-Intros; Unfold Rge; Elim (total_order r1 r2); Intro.
-Absurd ``r1<r2``; Trivial.
-Case H0; Auto.
-Qed.
-
-(**********)
-Lemma Rnot_lt_le:(r1,r2:R)~(``r1<r2``)->``r2<=r1``.
-Intros; Apply Rge_le; Apply Rlt_not_ge; Assumption.
-Qed.
-
-(**********)
-Lemma Rgt_not_le:(r1,r2:R)~(``r1>r2``)->``r1<=r2``.
-Intros r1 r2 H; Apply Rge_le.
-Exact (Rlt_not_ge r2 r1 H).
-Qed.
-
-(**********)
-Lemma Rgt_ge:(r1,r2:R)``r1>r2`` -> ``r1 >= r2``.
-Red; Auto with real.
-Qed.
-
-V7only [
-(**********)
-Lemma Rlt_sym:(r1,r2:R)``r1<r2`` <-> ``r2>r1``.
-Split; Unfold Rgt; Auto with real.
-Qed.
-
-(**********)
-Lemma Rle_sym1:(r1,r2:R)``r1<=r2``->``r2>=r1``.
-Proof Rle_ge.
-
-Notation "'Rle_sym2' a b" := (Rge_le b a)
- (at level 10, a,b at next level).
-Notation "'Rle_sym2' a" := [b:R](Rge_le b a)
- (at level 10, a at next level).
-Notation Rle_sym2 := Rge_le.
-(*
-(**********)
-Lemma Rle_sym2:(r1,r2:R)``r2>=r1`` -> ``r1<=r2``.
-Proof [r1,r2](Rge_le r2 r1).
-*)
-
-(**********)
-Lemma Rle_sym:(r1,r2:R)``r1<=r2``<->``r2>=r1``.
-Split; Auto with real.
-Qed.
-].
-
-(**********)
-Lemma Rge_gt_trans:(r1,r2,r3:R)``r1>=r2``->``r2>r3``->``r1>r3``.
-Unfold Rgt; Intros; Apply Rlt_le_trans with r2; Auto with real.
-Qed.
-
-(**********)
-Lemma Rgt_ge_trans:(r1,r2,r3:R)``r1>r2`` -> ``r2>=r3`` -> ``r1>r3``.
-Unfold Rgt; Intros; Apply Rle_lt_trans with r2; Auto with real.
-Qed.
-
-(**********)
-Lemma Rgt_trans:(r1,r2,r3:R)``r1>r2`` -> ``r2>r3`` -> ``r1>r3``.
-Unfold Rgt; Intros; Apply Rlt_trans with r2; Auto with real.
-Qed.
-
-(**********)
-Lemma Rge_trans:(r1,r2,r3:R)``r1>=r2`` -> ``r2>=r3`` -> ``r1>=r3``.
-Intros; Apply Rle_ge.
-Apply Rle_trans with r2; Auto with real.
-Qed.
-
-(**********)
-Lemma Rlt_r_plus_R1:(r:R)``0<=r`` -> ``0<r+1``.
-Intros.
-Apply Rlt_le_trans with ``1``; Auto with real.
-Pattern 1 ``1``; Replace ``1`` with ``0+1``; Auto with real.
-Qed.
-Hints Resolve Rlt_r_plus_R1: real.
-
-(**********)
-Lemma Rlt_r_r_plus_R1:(r:R)``r<r+1``.
-Intros.
-Pattern 1 r; Replace r with ``r+0``; Auto with real.
-Qed.
-Hints Resolve Rlt_r_r_plus_R1: real.
-
-(**********)
-Lemma tech_Rgt_minus:(r1,r2:R)``0<r2``->``r1>r1-r2``.
-Red; Unfold Rminus; Intros.
-Pattern 2 r1; Replace r1 with ``r1+0``; Auto with real.
-Qed.
-
-(***********)
-Lemma Rgt_plus_plus_r:(r,r1,r2:R)``r1>r2``->``r+r1 > r+r2``.
-Unfold Rgt; Auto with real.
-Qed.
-Hints Resolve Rgt_plus_plus_r : real.
-
-(***********)
-Lemma Rgt_r_plus_plus:(r,r1,r2:R)``r+r1 > r+r2`` -> ``r1 > r2``.
-Unfold Rgt; Intros; Apply (Rlt_anti_compatibility r r2 r1 H).
-Qed.
-
-(***********)
-Lemma Rge_plus_plus_r:(r,r1,r2:R)``r1>=r2`` -> ``r+r1 >= r+r2``.
-Intros; Apply Rle_ge; Auto with real.
-Qed.
-Hints Resolve Rge_plus_plus_r : real.
-
-(***********)
-Lemma Rge_r_plus_plus:(r,r1,r2:R)``r+r1 >= r+r2`` -> ``r1>=r2``.
-Intros; Apply Rle_ge; Apply Rle_anti_compatibility with r; Auto with real.
-Qed.
-
-(***********)
-Lemma Rmult_ge_compat_r:
- (z,x,y:R) ``z>=0`` -> ``x>=y`` -> ``x*z >= y*z``.
-Intros z x y; Intros; Apply Rle_ge; Apply Rle_monotony_r; Apply Rge_le; Assumption.
-Qed.
-
-V7only [
-Notation "'Rge_monotony' a b c" := (Rmult_ge_compat_r c a b)
- (at level 10, a,b,c at level 9, only parsing).
-Notation Rge_monotony := Rmult_ge_compat_r.
-].
-
-(***********)
-Lemma Rgt_minus:(r1,r2:R)``r1>r2`` -> ``r1-r2 > 0``.
-Intros; Replace ``0`` with ``r2-r2``; Auto with real.
-Unfold Rgt Rminus; Auto with real.
-Qed.
-
-(*********)
-Lemma minus_Rgt:(r1,r2:R)``r1-r2 > 0`` -> ``r1>r2``.
-Intros; Replace r2 with ``r2+0``; Auto with real.
-Intros; Replace r1 with ``r2+(r1-r2)``; Auto with real.
-Qed.
-
-(**********)
-Lemma Rge_minus:(r1,r2:R)``r1>=r2`` -> ``r1-r2 >= 0``.
-Unfold Rge; Intros; Elim H; Intro.
-Left; Apply (Rgt_minus r1 r2 H0).
-Right; Apply (eq_Rminus r1 r2 H0).
-Qed.
-
-(*********)
-Lemma minus_Rge:(r1,r2:R)``r1-r2 >= 0`` -> ``r1>=r2``.
-Intros; Replace r2 with ``r2+0``; Auto with real.
-Intros; Replace r1 with ``r2+(r1-r2)``; Auto with real.
-Qed.
-
-
-(*********)
-Lemma Rmult_gt:(r1,r2:R)``r1>0`` -> ``r2>0`` -> ``r1*r2>0``.
-Unfold Rgt;Intros.
-Replace ``0`` with ``0*r2``; Auto with real.
-Qed.
-
-(*********)
-Lemma Rmult_lt_pos:(x,y:R)``0<x`` -> ``0<y`` -> ``0<x*y``.
-Proof Rmult_gt.
-
-(***********)
-Lemma Rplus_eq_R0_l:(a,b:R)``0<=a`` -> ``0<=b`` -> ``a+b==0`` -> ``a==0``.
-Intros a b [H|H] H0 H1; Auto with real.
-Absurd ``0<a+b``.
-Rewrite H1; Auto with real.
-Replace ``0`` with ``0+0``; Auto with real.
-Qed.
-
-
-Lemma Rplus_eq_R0
- :(a,b:R)``0<=a`` -> ``0<=b`` -> ``a+b==0`` -> ``a==0``/\``b==0``.
-Intros a b; Split.
-Apply Rplus_eq_R0_l with b; Auto with real.
-Apply Rplus_eq_R0_l with a; Auto with real.
-Rewrite Rplus_sym; Auto with real.
-Qed.
-
-
-(***********)
-Lemma Rplus_Rsr_eq_R0_l:(a,b:R)``(Rsqr a)+(Rsqr b)==0``->``a==0``.
-Intros a b; Intros; Apply Rsqr_r_R0; Apply Rplus_eq_R0_l with (Rsqr b); Auto with real.
-Qed.
-
-Lemma Rplus_Rsr_eq_R0:(a,b:R)``(Rsqr a)+(Rsqr b)==0``->``a==0``/\``b==0``.
-Intros a b; Split.
-Apply Rplus_Rsr_eq_R0_l with b; Auto with real.
-Apply Rplus_Rsr_eq_R0_l with a; Auto with real.
-Rewrite Rplus_sym; Auto with real.
-Qed.
-
-
-(**********************************************************)
-(** Injection from [N] to [R] *)
-(**********************************************************)
-
-(**********)
-Lemma S_INR:(n:nat)(INR (S n))==``(INR n)+1``.
-Intro; Case n; Auto with real.
-Qed.
-
-(**********)
-Lemma S_O_plus_INR:(n:nat)
- (INR (plus (S O) n))==``(INR (S O))+(INR n)``.
-Intro; Simpl; Case n; Intros; Auto with real.
-Qed.
-
-(**********)
-Lemma plus_INR:(n,m:nat)(INR (plus n m))==``(INR n)+(INR m)``.
-Intros n m; Induction n.
-Simpl; Auto with real.
-Replace (plus (S n) m) with (S (plus n m)); Auto with arith.
-Repeat Rewrite S_INR.
-Rewrite Hrecn; Ring.
-Qed.
-
-(**********)
-Lemma minus_INR:(n,m:nat)(le m n)->(INR (minus n m))==``(INR n)-(INR m)``.
-Intros n m le; Pattern m n; Apply le_elim_rel; Auto with real.
-Intros; Rewrite <- minus_n_O; Auto with real.
-Intros; Repeat Rewrite S_INR; Simpl.
-Rewrite H0; Ring.
-Qed.
-
-(*********)
-Lemma mult_INR:(n,m:nat)(INR (mult n m))==(Rmult (INR n) (INR m)).
-Intros n m; Induction n.
-Simpl; Auto with real.
-Intros; Repeat Rewrite S_INR; Simpl.
-Rewrite plus_INR; Rewrite Hrecn; Ring.
-Qed.
-
-Hints Resolve plus_INR minus_INR mult_INR : real.
-
-(*********)
-Lemma lt_INR_0:(n:nat)(lt O n)->``0 < (INR n)``.
-Induction 1; Intros; Auto with real.
-Rewrite S_INR; Auto with real.
-Qed.
-Hints Resolve lt_INR_0: real.
-
-Lemma lt_INR:(n,m:nat)(lt n m)->``(INR n) < (INR m)``.
-Induction 1; Intros; Auto with real.
-Rewrite S_INR; Auto with real.
-Rewrite S_INR; Apply Rlt_trans with (INR m0); Auto with real.
-Qed.
-Hints Resolve lt_INR: real.
-
-Lemma INR_lt_1:(n:nat)(lt (S O) n)->``1 < (INR n)``.
-Intros;Replace ``1`` with (INR (S O));Auto with real.
-Qed.
-Hints Resolve INR_lt_1: real.
-
-(**********)
-Lemma INR_pos : (p:positive)``0<(INR (convert p))``.
-Intro; Apply lt_INR_0.
-Simpl; Auto with real.
-Apply compare_convert_O.
-Qed.
-Hints Resolve INR_pos : real.
-
-(**********)
-Lemma pos_INR:(n:nat)``0 <= (INR n)``.
-Intro n; Case n.
-Simpl; Auto with real.
-Auto with arith real.
-Qed.
-Hints Resolve pos_INR: real.
-
-Lemma INR_lt:(n,m:nat)``(INR n) < (INR m)``->(lt n m).
-Double Induction n m;Intros.
-Simpl;ElimType False;Apply (Rlt_antirefl R0);Auto.
-Auto with arith.
-Generalize (pos_INR (S n0));Intro;Cut (INR O)==R0;
- [Intro H2;Rewrite H2 in H0;Idtac|Simpl;Trivial].
-Generalize (Rle_lt_trans ``0`` (INR (S n0)) ``0`` H1 H0);Intro;
- ElimType False;Apply (Rlt_antirefl R0);Auto.
-Do 2 Rewrite S_INR in H1;Cut ``(INR n1) < (INR n0)``.
-Intro H2;Generalize (H0 n0 H2);Intro;Auto with arith.
-Apply (Rlt_anti_compatibility ``1`` (INR n1) (INR n0)).
-Rewrite Rplus_sym;Rewrite (Rplus_sym ``1`` (INR n0));Trivial.
-Qed.
-Hints Resolve INR_lt: real.
-
-(*********)
-Lemma le_INR:(n,m:nat)(le n m)->``(INR n)<=(INR m)``.
-Induction 1; Intros; Auto with real.
-Rewrite S_INR.
-Apply Rle_trans with (INR m0); Auto with real.
-Qed.
-Hints Resolve le_INR: real.
-
-(**********)
-Lemma not_INR_O:(n:nat)``(INR n)<>0``->~n=O.
-Red; Intros n H H1.
-Apply H.
-Rewrite H1; Trivial.
-Qed.
-Hints Immediate not_INR_O : real.
-
-(**********)
-Lemma not_O_INR:(n:nat)~n=O->``(INR n)<>0``.
-Intro n; Case n.
-Intro; Absurd (0)=(0); Trivial.
-Intros; Rewrite S_INR.
-Apply Rgt_not_eq; Red; Auto with real.
-Qed.
-Hints Resolve not_O_INR : real.
-
-Lemma not_nm_INR:(n,m:nat)~n=m->``(INR n)<>(INR m)``.
-Intros n m H; Case (le_or_lt n m); Intros H1.
-Case (le_lt_or_eq ? ? H1); Intros H2.
-Apply imp_not_Req; Auto with real.
-ElimType False;Auto.
-Apply sym_not_eqT; Apply imp_not_Req; Auto with real.
-Qed.
-Hints Resolve not_nm_INR : real.
-
-Lemma INR_eq: (n,m:nat)(INR n)==(INR m)->n=m.
-Intros;Case (le_or_lt n m); Intros H1.
-Case (le_lt_or_eq ? ? H1); Intros H2;Auto.
-Cut ~n=m.
-Intro H3;Generalize (not_nm_INR n m H3);Intro H4;
- ElimType False;Auto.
-Omega.
-Symmetry;Cut ~m=n.
-Intro H3;Generalize (not_nm_INR m n H3);Intro H4;
- ElimType False;Auto.
-Omega.
-Qed.
-Hints Resolve INR_eq : real.
-
-Lemma INR_le: (n, m : nat) (Rle (INR n) (INR m)) -> (le n m).
-Intros;Elim H;Intro.
-Generalize (INR_lt n m H0);Intro;Auto with arith.
-Generalize (INR_eq n m H0);Intro;Rewrite H1;Auto.
-Qed.
-Hints Resolve INR_le : real.
-
-Lemma not_1_INR:(n:nat)~n=(S O)->``(INR n)<>1``.
-Replace ``1`` with (INR (S O)); Auto with real.
-Qed.
-Hints Resolve not_1_INR : real.
-
-(**********************************************************)
-(** Injection from [Z] to [R] *)
-(**********************************************************)
-
-V7only [
-(**********)
-Definition Z_of_nat := inject_nat.
-Notation INZ:=Z_of_nat.
-].
-
-(**********)
-Lemma IZN:(z:Z)(`0<=z`)->(Ex [m:nat] z=(INZ m)).
-Intros z; Unfold INZ; Apply inject_nat_complete; Assumption.
-Qed.
-
-(**********)
-Lemma INR_IZR_INZ:(n:nat)(INR n)==(IZR (INZ n)).
-Induction n; Auto with real.
-Intros; Simpl; Rewrite bij1; Auto with real.
-Qed.
-
-Lemma plus_IZR_NEG_POS :
- (p,q:positive)(IZR `(POS p)+(NEG q)`)==``(IZR (POS p))+(IZR (NEG q))``.
-Intros.
-Case (lt_eq_lt_dec (convert p) (convert q)).
-Intros [H | H]; Simpl.
-Rewrite convert_compare_INFERIEUR; Simpl; Trivial.
-Rewrite (true_sub_convert q p).
-Rewrite minus_INR; Auto with arith; Ring.
-Apply ZC2; Apply convert_compare_INFERIEUR; Trivial.
-Rewrite (convert_intro p q); Trivial.
-Rewrite convert_compare_EGAL; Simpl; Auto with real.
-Intro H; Simpl.
-Rewrite convert_compare_SUPERIEUR; Simpl; Auto with arith.
-Rewrite (true_sub_convert p q).
-Rewrite minus_INR; Auto with arith; Ring.
-Apply ZC2; Apply convert_compare_INFERIEUR; Trivial.
-Qed.
-
-(**********)
-Lemma plus_IZR:(z,t:Z)(IZR `z+t`)==``(IZR z)+(IZR t)``.
-Intro z; NewDestruct z; Intro t; NewDestruct t; Intros; Auto with real.
-Simpl; Intros; Rewrite convert_add; Auto with real.
-Apply plus_IZR_NEG_POS.
-Rewrite Zplus_sym; Rewrite Rplus_sym; Apply plus_IZR_NEG_POS.
-Simpl; Intros; Rewrite convert_add; Rewrite plus_INR; Auto with real.
-Qed.
-
-(**********)
-Lemma mult_IZR:(z,t:Z)(IZR `z*t`)==``(IZR z)*(IZR t)``.
-Intros z t; Case z; Case t; Simpl; Auto with real.
-Intros t1 z1; Rewrite times_convert; Auto with real.
-Intros t1 z1; Rewrite times_convert; Auto with real.
-Rewrite Rmult_sym.
-Rewrite Ropp_mul1; Auto with real.
-Apply eq_Ropp; Rewrite mult_sym; Auto with real.
-Intros t1 z1; Rewrite times_convert; Auto with real.
-Rewrite Ropp_mul1; Auto with real.
-Intros t1 z1; Rewrite times_convert; Auto with real.
-Rewrite Ropp_mul2; Auto with real.
-Qed.
-
-(**********)
-Lemma Ropp_Ropp_IZR:(z:Z)(IZR (`-z`))==``-(IZR z)``.
-Intro z; Case z; Simpl; Auto with real.
-Qed.
-
-(**********)
-Lemma Z_R_minus:(z1,z2:Z)``(IZR z1)-(IZR z2)``==(IZR `z1-z2`).
-Intros z1 z2; Unfold Rminus; Unfold Zminus.
-Rewrite <-(Ropp_Ropp_IZR z2); Symmetry; Apply plus_IZR.
-Qed.
-
-(**********)
-Lemma lt_O_IZR:(z:Z)``0 < (IZR z)``->`0<z`.
-Intro z; Case z; Simpl; Intros.
-Absurd ``0<0``; Auto with real.
-Unfold Zlt; Simpl; Trivial.
-Case Rlt_le_not with 1:=H.
-Replace ``0`` with ``-0``; Auto with real.
-Qed.
-
-(**********)
-Lemma lt_IZR:(z1,z2:Z)``(IZR z1)<(IZR z2)``->`z1<z2`.
-Intros z1 z2 H; Apply Zlt_O_minus_lt.
-Apply lt_O_IZR.
-Rewrite <- Z_R_minus.
-Exact (Rgt_minus (IZR z2) (IZR z1) H).
-Qed.
-
-(**********)
-Lemma eq_IZR_R0:(z:Z)``(IZR z)==0``->`z=0`.
-Intro z; NewDestruct z; Simpl; Intros; Auto with zarith.
-Case (Rlt_not_eq ``0`` (INR (convert p))); Auto with real.
-Case (Rlt_not_eq ``-(INR (convert p))`` ``0`` ); Auto with real.
-Apply Rgt_RoppO. Unfold Rgt; Apply INR_pos.
-Qed.
-
-(**********)
-Lemma eq_IZR:(z1,z2:Z)(IZR z1)==(IZR z2)->z1=z2.
-Intros z1 z2 H;Generalize (eq_Rminus (IZR z1) (IZR z2) H);
- Rewrite (Z_R_minus z1 z2);Intro;Generalize (eq_IZR_R0 `z1-z2` H0);
- Intro;Omega.
-Qed.
-
-(**********)
-Lemma not_O_IZR:(z:Z)`z<>0`->``(IZR z)<>0``.
-Intros z H; Red; Intros H0; Case H.
-Apply eq_IZR; Auto.
-Qed.
-
-(*********)
-Lemma le_O_IZR:(z:Z)``0<= (IZR z)``->`0<=z`.
-Unfold Rle; Intros z [H|H].
-Red;Intro;Apply (Zlt_le_weak `0` z (lt_O_IZR z H)); Assumption.
-Rewrite (eq_IZR_R0 z); Auto with zarith real.
-Qed.
-
-(**********)
-Lemma le_IZR:(z1,z2:Z)``(IZR z1)<=(IZR z2)``->`z1<=z2`.
-Unfold Rle; Intros z1 z2 [H|H].
-Apply (Zlt_le_weak z1 z2); Auto with real.
-Apply lt_IZR; Trivial.
-Rewrite (eq_IZR z1 z2); Auto with zarith real.
-Qed.
-
-(**********)
-Lemma le_IZR_R1:(z:Z)``(IZR z)<=1``-> `z<=1`.
-Pattern 1 ``1``; Replace ``1`` with (IZR `1`); Intros; Auto.
-Apply le_IZR; Trivial.
-Qed.
-
-(**********)
-Lemma IZR_ge: (m,n:Z) `m>= n` -> ``(IZR m)>=(IZR n)``.
-Intros m n H; Apply Rlt_not_ge;Red;Intro.
-Generalize (lt_IZR m n H0); Intro; Omega.
-Qed.
-
-Lemma IZR_le: (m,n:Z) `m<= n` -> ``(IZR m)<=(IZR n)``.
-Intros m n H;Apply Rgt_not_le;Red;Intro.
-Unfold Rgt in H0;Generalize (lt_IZR n m H0); Intro; Omega.
-Qed.
-
-Lemma IZR_lt: (m,n:Z) `m< n` -> ``(IZR m)<(IZR n)``.
-Intros m n H;Cut `m<=n`.
-Intro H0;Elim (IZR_le m n H0);Intro;Auto.
-Generalize (eq_IZR m n H1);Intro;ElimType False;Omega.
-Omega.
-Qed.
-
-Lemma one_IZR_lt1 : (z:Z)``-1<(IZR z)<1``->`z=0`.
-Intros z (H1,H2).
-Apply Zle_antisym.
-Apply Zlt_n_Sm_le; Apply lt_IZR; Trivial.
-Replace `0` with (Zs `-1`); Trivial.
-Apply Zlt_le_S; Apply lt_IZR; Trivial.
-Qed.
-
-Lemma one_IZR_r_R1
- : (r:R)(z,x:Z)``r<(IZR z)<=r+1``->``r<(IZR x)<=r+1``->z=x.
-Intros r z x (H1,H2) (H3,H4).
-Cut `z-x=0`; Auto with zarith.
-Apply one_IZR_lt1.
-Rewrite <- Z_R_minus; Split.
-Replace ``-1`` with ``r-(r+1)``.
-Unfold Rminus; Apply Rplus_lt_le_lt; Auto with real.
-Ring.
-Replace ``1`` with ``(r+1)-r``.
-Unfold Rminus; Apply Rplus_le_lt_lt; Auto with real.
-Ring.
-Qed.
-
-
-(**********)
-Lemma single_z_r_R1:
- (r:R)(z,x:Z)``r<(IZR z)``->``(IZR z)<=r+1``->``r<(IZR x)``->
- ``(IZR x)<=r+1``->z=x.
-Intros; Apply one_IZR_r_R1 with r; Auto.
-Qed.
-
-(**********)
-Lemma tech_single_z_r_R1
- :(r:R)(z:Z)``r<(IZR z)``->``(IZR z)<=r+1``
- -> (Ex [s:Z] (~s=z/\``r<(IZR s)``/\``(IZR s)<=r+1``))->False.
-Intros r z H1 H2 (s, (H3,(H4,H5))).
-Apply H3; Apply single_z_r_R1 with r; Trivial.
-Qed.
-
-(*****************************************************************)
-(** Definitions of new types *)
-(*****************************************************************)
-
-Record nonnegreal : Type := mknonnegreal {
-nonneg :> R;
-cond_nonneg : ``0<=nonneg`` }.
-
-Record posreal : Type := mkposreal {
-pos :> R;
-cond_pos : ``0<pos`` }.
-
-Record nonposreal : Type := mknonposreal {
-nonpos :> R;
-cond_nonpos : ``nonpos<=0`` }.
-
-Record negreal : Type := mknegreal {
-neg :> R;
-cond_neg : ``neg<0`` }.
-
-Record nonzeroreal : Type := mknonzeroreal {
-nonzero :> R;
-cond_nonzero : ~``nonzero==0`` }.
-
-(**********)
-Lemma prod_neq_R0 : (x,y:R) ~``x==0``->~``y==0``->~``x*y==0``.
-Intros x y; Intros; Red; Intro; Generalize (without_div_Od x y H1); Intro; Elim H2; Intro; [Rewrite H3 in H; Elim H | Rewrite H3 in H0; Elim H0]; Reflexivity.
-Qed.
-
-(*********)
-Lemma Rmult_le_pos : (x,y:R) ``0<=x`` -> ``0<=y`` -> ``0<=x*y``.
-Intros x y H H0; Rewrite <- (Rmult_Ol x); Rewrite <- (Rmult_sym x); Apply (Rle_monotony x R0 y H H0).
-Qed.
-
-Lemma double : (x:R) ``2*x==x+x``.
-Intro; Ring.
-Qed.
-
-Lemma double_var : (x:R) ``x == x/2 + x/2``.
-Intro; Rewrite <- double; Unfold Rdiv; Rewrite <- Rmult_assoc; Symmetry; Apply Rinv_r_simpl_m.
-Replace ``2`` with (INR (2)); [Apply not_O_INR; Discriminate | Unfold INR; Ring].
-Qed.
-
-(**********************************************************)
-(** Other rules about < and <= *)
-(**********************************************************)
-
-Lemma gt0_plus_gt0_is_gt0 : (x,y:R) ``0<x`` -> ``0<y`` -> ``0<x+y``.
-Intros x y; Intros; Apply Rlt_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rlt_compatibility; Assumption].
-Qed.
-
-Lemma ge0_plus_gt0_is_gt0 : (x,y:R) ``0<=x`` -> ``0<y`` -> ``0<x+y``.
-Intros x y; Intros; Apply Rle_lt_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rlt_compatibility; Assumption].
-Qed.
-
-Lemma gt0_plus_ge0_is_gt0 : (x,y:R) ``0<x`` -> ``0<=y`` -> ``0<x+y``.
-Intros x y; Intros; Rewrite <- Rplus_sym; Apply ge0_plus_gt0_is_gt0; Assumption.
-Qed.
-
-Lemma ge0_plus_ge0_is_ge0 : (x,y:R) ``0<=x`` -> ``0<=y`` -> ``0<=x+y``.
-Intros x y; Intros; Apply Rle_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption].
-Qed.
-
-Lemma plus_le_is_le : (x,y,z:R) ``0<=y`` -> ``x+y<=z`` -> ``x<=z``.
-Intros x y z; Intros; Apply Rle_trans with ``x+y``; [Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption | Assumption].
-Qed.
-
-Lemma plus_lt_is_lt : (x,y,z:R) ``0<=y`` -> ``x+y<z`` -> ``x<z``.
-Intros x y z; Intros; Apply Rle_lt_trans with ``x+y``; [Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption | Assumption].
-Qed.
-
-Lemma Rmult_lt2 : (r1,r2,r3,r4:R) ``0<=r1`` -> ``0<=r3`` -> ``r1<r2`` -> ``r3<r4`` -> ``r1*r3<r2*r4``.
-Intros; Apply Rle_lt_trans with ``r2*r3``; [Apply Rle_monotony_r; [Assumption | Left; Assumption] | Apply Rlt_monotony; [Apply Rle_lt_trans with r1; Assumption | Assumption]].
-Qed.
-
-Lemma le_epsilon : (x,y:R) ((eps : R) ``0<eps``->``x<=y+eps``) -> ``x<=y``.
-Intros x y; Intros; Elim (total_order x y); Intro.
-Left; Assumption.
-Elim H0; Intro.
-Right; Assumption.
-Clear H0; Generalize (Rgt_minus x y H1); Intro H2; Change ``0<x-y`` in H2.
-Cut ``0<2``.
-Intro.
-Generalize (Rmult_lt_pos ``x-y`` ``/2`` H2 (Rlt_Rinv ``2`` H0)); Intro H3; Generalize (H ``(x-y)*/2`` H3); Replace ``y+(x-y)*/2`` with ``(y+x)*/2``.
-Intro H4; Generalize (Rle_monotony ``2`` x ``(y+x)*/2`` (Rlt_le ``0`` ``2`` H0) H4); Rewrite <- (Rmult_sym ``((y+x)*/2)``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym.
-Rewrite Rmult_1r; Replace ``2*x`` with ``x+x``.
-Rewrite (Rplus_sym y); Intro H5; Apply Rle_anti_compatibility with x; Assumption.
-Ring.
-Replace ``2`` with (INR (S (S O))); [Apply not_O_INR; Discriminate | Ring].
-Pattern 2 y; Replace y with ``y/2+y/2``.
-Unfold Rminus Rdiv.
-Repeat Rewrite Rmult_Rplus_distrl.
-Ring.
-Cut (z:R) ``2*z == z + z``.
-Intro.
-Rewrite <- (H4 ``y/2``).
-Unfold Rdiv.
-Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m.
-Replace ``2`` with (INR (2)).
-Apply not_O_INR.
-Discriminate.
-Unfold INR; Reflexivity.
-Intro; Ring.
-Cut ~(O=(2)); [Intro H0; Generalize (lt_INR_0 (2) (neq_O_lt (2) H0)); Unfold INR; Intro; Assumption | Discriminate].
-Qed.
-
-(**********)
-Lemma complet_weak : (E:R->Prop) (bound E) -> (ExT [x:R] (E x)) -> (ExT [m:R] (is_lub E m)).
-Intros; Elim (complet E H H0); Intros; Split with x; Assumption.
-Qed.
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