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diff --git a/theories7/Reals/RIneq.v b/theories7/Reals/RIneq.v new file mode 100644 index 00000000..00d41c70 --- /dev/null +++ b/theories7/Reals/RIneq.v @@ -0,0 +1,1631 @@ +(************************************************************************) +(* 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. |