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author | 2001-12-05 17:29:31 +0000 | |
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committer | 2001-12-05 17:29:31 +0000 | |
commit | 61002f54b95e5d947e48fa8df22bb5758df4422f (patch) | |
tree | 889f0a4a3864c84db1e723946498e8af837a8bfb /theories/Reals/Ranalysis.v | |
parent | d1e24b490ff371b83c9c9a53313ebf3674f08790 (diff) |
*** empty log message ***
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2274 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Reals/Ranalysis.v')
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1 files changed, 808 insertions, 0 deletions
diff --git a/theories/Reals/Ranalysis.v b/theories/Reals/Ranalysis.v new file mode 100644 index 000000000..fb5ec988e --- /dev/null +++ b/theories/Reals/Ranalysis.v @@ -0,0 +1,808 @@ +Require Rbase. +Require Rbasic_fun. +Require R_sqr. +Require Rlimit. +Require Rderiv. +Require DiscrR. +Require Rtrigo. + +(****************************************************) +(* Basic operations on functions *) +(****************************************************) +Definition plus_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)+(f2 x)``. +Definition opp_fct [f:R->R] : R->R := [x:R] ``-(f x)``. +Definition mult_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)*(f2 x)``. +Definition mult_real_fct [a:R;f:R->R] : R->R := [x:R] ``a*(f x)``. +Definition minus_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)-(f2 x)``. +Definition div_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)/(f2 x)``. +Definition div_real_fct [a:R;f:R->R] : R->R := [x:R] ``a/(f x)``. +Definition comp [f1,f2:R->R] : R->R := [x:R] ``(f1 (f2 x))``. + +(****************************************************) +(* Variations of functions *) +(****************************************************) +Definition increasing [f:R->R] : Prop := (x,y:R) ``x<=y``->``(f x)<=(f y)``. +Definition decreasing [f:R->R] : Prop := (x,y:R) ``x<=y``->``(f y)<=(f x)``. +Definition strict_increasing [f:R->R] : Prop := (x,y:R) ``x<y``->``(f x)<(f y)``. +Definition strict_decreasing [f:R->R] : Prop := (x,y:R) ``x<y``->``(f y)<(f x)``. +Definition constant [f:R->R] : Prop := (x,y:R) ``(f x)==(f y)``. + +(**********) +Axiom fct_eq : (A,B:Type) (f1,f2:A->B) ((x:A)(f1 x)==(f2 x))->f1==f2. + +(**********) +Definition no_cond : R->Prop := [x:R] True. + +(***************************************************) +(* Definition of continuity as a limit *) +(***************************************************) + +(**********) +Definition continuity_pt [f:R->R; x0:R] : Prop := (continue_in f no_cond x0). + +(**********) +Lemma sum_continuous : (f1,f2:R->R; x0:R) (continuity_pt f1 x0) -> (continuity_pt f2 x0) -> (continuity_pt (plus_fct f1 f2) x0). +Unfold continuity_pt plus_fct; Unfold continue_in; Intros; Apply limit_plus; Assumption. +Save. + +(**********) +Lemma diff_continuous : (f1,f2:R->R; x0:R) (continuity_pt f1 x0) -> (continuity_pt f2 x0) -> (continuity_pt (minus_fct f1 f2) x0). +Unfold continuity_pt minus_fct; Unfold continue_in; Intros; Apply limit_minus; Assumption. +Save. + +(**********) +Lemma prod_continuous : (f1,f2:R->R; x0:R) (continuity_pt f1 x0) -> (continuity_pt f2 x0) -> (continuity_pt (mult_fct f1 f2) x0). +Unfold continuity_pt mult_fct; Unfold continue_in; Intros; Apply limit_mul; Assumption. +Save. + +(**********) +Lemma const_continuous : (f:R->R; x0:R) (constant f) -> (continuity_pt f x0). +Unfold constant continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Intros; Exists ``1``; Split; [Apply Rlt_R0_R1 | Intros; Generalize (H x x0); Intro; Rewrite H2; Simpl; Rewrite R_dist_eq; Assumption]. +Save. + +(**********) +Lemma scal_continuous : (f:R->R;a:R; x0:R) (continuity_pt f x0) -> (continuity_pt (mult_real_fct a f) x0). +Unfold continuity_pt mult_real_fct; Unfold continue_in; Intros; Apply (limit_mul ([x:R] a) f (D_x no_cond x0) a (f x0) x0). +Unfold limit1_in; Unfold limit_in; Intros; Exists ``1``; Split. +Apply Rlt_R0_R1. +Intros; Rewrite R_dist_eq; Assumption. +Assumption. +Save. + +(**********) +Lemma opp_continuous : (f:R->R; x0:R) (continuity_pt f x0) -> (continuity_pt (opp_fct f) x0). +Unfold continuity_pt opp_fct; Unfold continue_in; Intros; Apply limit_Ropp; Assumption. +Save. + +(**********) +Lemma inv_continuous : (f:R->R; x0:R) (continuity_pt f x0) -> ~``(f x0)==0`` -> +(continuity_pt ([x:R] ``/(f x)``) x0). +Unfold continuity_pt; Unfold continue_in; Intros; Apply limit_inv; Assumption. +Save. + +Lemma div_eq_inv : (f1,f2:R->R) (div_fct f1 f2)==(mult_fct f1 ([x:R]``/(f2 x)``)). +Intros; Unfold div_fct; Unfold mult_fct; Unfold Rdiv; Apply fct_eq; Intro x; Reflexivity. +Save. + +(**********) +Lemma div_continuous : (f1,f2:R->R; x0:R) (continuity_pt f1 x0) -> (continuity_pt f2 x0) -> ~``(f2 x0)==0`` -> (continuity_pt (div_fct f1 f2) x0). +Intros; Rewrite -> (div_eq_inv f1 f2); Apply prod_continuous; [Assumption | Apply inv_continuous; Assumption]. +Save. + +(**********) +Definition continuity [f:R->R] : Prop := (x:R) (continuity_pt f x). + +Lemma sum_continuity : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (plus_fct f1 f2)). +Unfold continuity; Intros; Apply (sum_continuous f1 f2 x (H x) (H0 x)). +Save. + +Lemma diff_continuity : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (minus_fct f1 f2)). +Unfold continuity; Intros; Apply (diff_continuous f1 f2 x (H x) (H0 x)). +Save. + +Lemma prod_continuity : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (mult_fct f1 f2)). +Unfold continuity; Intros; Apply (prod_continuous f1 f2 x (H x) (H0 x)). +Save. + +Lemma const_continuity : (f:R->R) (constant f) -> (continuity f). +Unfold continuity; Intros; Apply (const_continuous f x H). +Save. + +Lemma scal_continuity : (f:R->R;a:R) (continuity f) -> (continuity (mult_real_fct a f)). +Unfold continuity; Intros; Apply (scal_continuous f a x (H x)). +Save. + +Lemma opp_continuity : (f:R->R) (continuity f)->(continuity (opp_fct f)). +Unfold continuity; Intros; Apply (opp_continuous f x (H x)). +Save. + +Lemma div_continuity : (f1,f2:R->R) (continuity f1)->(continuity f2)->((x:R) ~``(f2 x)==0``)->(continuity (div_fct f1 f2)). +Unfold continuity; Intros; Apply (div_continuous f1 f2 x (H x) (H0 x) (H1 x)). +Save. + +Lemma inv_continuity : (f:R->R) (continuity f)->((x:R) ~``(f x)==0``)->(continuity ([x:R] ``/(f x)``)). +Unfold continuity; Intros; Apply (inv_continuous f x (H x) (H0 x)). +Save. + +(*****************************************************) +(* Derivative's definition using Landau's kernel *) +(*****************************************************) +Definition derivable_pt [f:R->R; x:R] : Prop := (EXT l : R | ((eps:R) ``0<eps``->(EXT delta : posreal | ((h:R) ~``h==0``->``(Rabsolu h)<delta`` -> ``(Rabsolu ((((f (x+h))-(f x))/h)-l))<eps``)))). + +Definition derivable [f:R->R] : Prop := (x:R) (derivable_pt f x). + +Parameter derive_pt : (R->R)->R->R. + +Axiom derive_pt_def : (f:R->R;x,l:R) ((eps:R) ``0<eps``->(EXT delta : posreal | ((h:R) ~``h==0``->``(Rabsolu h)<delta`` -> ``(Rabsolu ((((f (x+h))-(f x))/h)-l))<eps``))) <-> (derive_pt f x)==l. + +(**********) +Lemma derive_pt_def_0 : (f:R->R;x,l:R) ((eps:R) ``0<eps``->(EXT delta : posreal | ((h:R) ~``h==0``->``(Rabsolu h)<delta`` -> ``(Rabsolu ((((f (x+h))-(f x))/h)-l))<eps``))) -> (derive_pt f x)==l. +Intros; Elim (derive_pt_def f x l); Intros; Apply (H0 H). +Save. + +(**********) +Lemma derive_pt_def_1 : (f:R->R;x,l:R) (derive_pt f x)==l -> ((eps:R) ``0<eps``->(EXT delta : posreal | ((h:R) ~``h==0``->``(Rabsolu h)<delta`` -> ``(Rabsolu ((((f (x+h))-(f x))/h)-l))<eps``))). +Intros; Elim (derive_pt_def f x l); Intros; Apply (H2 H eps H0). +Save. + +(**********) +Definition derive [f:R->R] := [x:R] (derive_pt f x). + +(***********************************) +(* Class of differential functions *) +(***********************************) +Record Differential : Type := mkDifferential { +d1 :> R->R; +cond_diff : (derivable d1) }. + +Record Differential_D2 : Type := mkDifferential_D2 { +d2 :> R->R; +cond_D1 : (derivable d2); +cond_D2 : (derivable (derive d2)) }. + +(**********) +Lemma derivable_derive : (f:R->R;x:R) (derivable_pt f x) -> (EXT l : R | (derive_pt f x)==l). +Intros f x; Unfold derivable_pt; Intro H; Elim H; Intros l H0; Rewrite (derive_pt_def_0 f x l); [Exists l; Reflexivity | Assumption]. +Save. + +(**********) +Lemma derive_derivable : (f:R->R;x,l:R) (derive_pt f x)==l -> (derivable_pt f x). +Intros; Unfold derivable_pt; Generalize (derive_pt_def_1 f x l H); Intro H0; Exists l; Assumption. +Save. + +(*******************************************************************) +(* Equivalence of this definition with the one using limit concept *) +(*******************************************************************) +Lemma derive_pt_D_in : (f,df:R->R;x:R) (D_in f df no_cond x) <-> (derive_pt +f x)==(df x). +Intros; Split. +Unfold D_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. +Apply derive_pt_def_0. +Intros; Elim (H eps H0); Intros alpha H1; Elim H1; Intros; Exists (mkposreal alpha H2); Intros; Generalize (H3 ``x+h``); Intro; Cut ``x+h-x==h``; [Intro; Cut ``(D_x no_cond x (x+h))``/\``(Rabsolu (x+h-x)) < alpha``; [Intro; Generalize (H6 H8); Rewrite H7; Intro; Assumption | Split; [Unfold D_x; Split; [Unfold no_cond; Trivial | Apply Rminus_not_eq_right; Rewrite H7; Assumption] | Rewrite H7; Assumption]] | Ring]. +Intro; Generalize (derive_pt_def_1 f x (df x) H); Intro; Unfold D_in; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros; Elim (H0 eps H1); Intros alpha H2; Exists (pos alpha); Split. +Apply (cond_pos alpha). +Intros; Elim H3; Intros; Unfold D_x in H4; Elim H4; Intros; Cut ``x0-x<>0``. +Intro; Generalize (H2 ``x0-x`` H8 H5); Replace ``x+(x0-x)`` with x0. +Intro; Assumption. +Ring. +Auto with real. +Save. + +Definition fct_cte [a:R] : R->R := [x:R]a. + +(***********************************) +(* derivability -> continuity *) +(***********************************) +Theorem derivable_continuous_pt : (f:R->R;x:R) (derivable_pt f x) -> (continuity_pt f x). +Intros. +Generalize (derivable_derive f x H); Intro. +Elim H0; Intros l H1. +Cut l==((fct_cte l) x). +Intro. +Rewrite H2 in H1. +Generalize (derive_pt_D_in f (fct_cte l) x); Intro. +Elim H3; Intros. +Generalize (H5 H1); Intro. +Unfold continuity_pt. +Apply (cont_deriv f (fct_cte l) no_cond x H6). +Unfold fct_cte; Reflexivity. +Save. + +Theorem derivable_continuous : (f:R->R) (derivable f) -> (continuity f). +Unfold derivable continuity; Intros; Apply (derivable_continuous_pt f x (H x)). +Save. + +(****************************************************************) +(* Main rules *) +(****************************************************************) + +(* Addition *) +Lemma deriv_sum : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> ``(derive_pt (plus_fct f1 f2) x)==(derive_pt f1 x)+(derive_pt f2 x)``. +Intros; Generalize (derivable_derive f1 x H); Intro H1; Generalize (derivable_derive f2 x H0); Intro H2; Elim H1; Clear H1; Intros l1 H1; Elim H2; Clear H2; Intros l2 H2; Unfold plus_fct; Rewrite H1; Rewrite H2; Apply derive_pt_def_0; Intros; Generalize (derive_pt_def_1 f1 x l1 H1); Clear H1; Intro H1; Generalize (derive_pt_def_1 f2 x l2 H2); Clear H2; Intro H2; Cut ~(O=(2)). +Intro Haux; Generalize (lt_INR_0 (2) (neq_O_lt (2) Haux)); Rewrite INR_eq_INR2; Unfold INR2; Intro Haux1; Generalize (Rlt_Rinv ``2`` Haux1); Clear Haux1; Intro Haux1; Generalize (Rmult_lt_pos eps ``/2`` H3 Haux1); Clear Haux1; Intro Haux1; Elim (H1 ``eps/2`` Haux1); Intros delta1 H4; Elim (H2 ``eps/2`` Haux1); Intros delta2 H5; Exists (mkposreal (Rmin delta1 delta2) (Rmin_stable_in_posreal delta1 delta2)); Intros h H6 H7; Unfold plus_fct; Replace ``((f1 (x+h))+(f2 (x+h))-((f1 x)+(f2 x)))/h-(l1+l2)`` with ``(((f1 (x+h))-(f1 x))/h-l1)+(((f2 (x+h))-(f2 x))/h-l2)``. +Apply Rle_lt_trans with ``(Rabsolu ((f1 (x+h))-(f1 x))/h-l1)+(Rabsolu ((f2 (x+h))-(f2 x))/h-l2)``. +Apply Rabsolu_triang. +Generalize (H5 h H6 (Rlt_le_trans (Rabsolu h) (Rmin delta1 delta2) delta2 H7 (Rmin_r delta1 delta2))); Intro H8; Generalize (H4 h H6 (Rlt_le_trans (Rabsolu h) (Rmin delta1 delta2) delta1 H7 (Rmin_l delta1 delta2))); Intro H9. +Generalize (Rplus_lt ``(Rabsolu (((f1 (x+h))-(f1 x))/h-l1))`` ``eps/2`` ``(Rabsolu (((f2 (x+h))-(f2 x))/h-l2))`` ``eps/2`` H9 H8). +Replace ``eps/2+eps/2`` with ``eps``. +Intro H10; Assumption. +Field; DiscrR. +Field; Assumption. +Discriminate. +Save. + +Lemma sum_derivable_pt : (f1,f2:R->R;x:R) (derivable_pt f1 x)->(derivable_pt f2 x)->(derivable_pt (plus_fct f1 f2) x). +Intros; Generalize (derivable_derive f1 x H); Intro; Generalize (derivable_derive f2 x H0); Intro; Elim H1; Clear H1; Intros l1 H1; Elim H2; Clear H2; Intros l2 H2; Apply (derive_derivable (plus_fct f1 f2) x ``l1+l2``); Rewrite <- H1; Rewrite <- H2; Apply deriv_sum; Assumption. +Save. + +Lemma sum_derivable : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (plus_fct f1 f2)). +Unfold derivable; Intros f1 f2 H1 H2 x; Apply sum_derivable_pt; [Exact (H1 x) | Exact (H2 x)]. +Save. + +(* Opposite *) +Lemma deriv_opposite : (f:R->R;x:R) (derivable_pt f x) -> ``(derive_pt (opp_fct f) x)==-(derive_pt f x)``. +Intros; Generalize (derivable_derive f x H); Intro H0; Elim H0; Intros l H1; Rewrite H1; Unfold opp_fct; Apply derive_pt_def_0; Intros; Generalize (derive_pt_def_1 f x l H1); Intro H3; Elim (H3 eps H2); Intros delta H4; Exists delta; Intros; Replace ``( -(f (x+h))- -(f x))/h- -l`` with ``- (((f (x+h))-(f x))/h-l)``; [Rewrite Rabsolu_Ropp; Apply (H4 h H5 H6) | Field; Assumption]. +Save. + +Lemma opposite_derivable_pt : (f:R->R;x:R) (derivable_pt f x) -> (derivable_pt (opp_fct f) x). +Unfold opp_fct derivable_pt; Intros; Elim H; Intros; Exists ``-x0``; Intros; Elim (H0 eps H1); Intros; Exists x1; Intros; Generalize (H2 h H3 H4); Intro H5; Replace ``( -(f (x+h))- -(f x))/h- -x0`` with ``- (((f (x+h))-(f x))/h-x0)``; [Rewrite Rabsolu_Ropp; Assumption | Field; Assumption]. +Save. + +Lemma opposite_derivable : (f:R->R) (derivable f) -> (derivable (opp_fct f)). +Unfold derivable; Intros f H1 x; Apply opposite_derivable_pt; Exact (H1 x). +Save. + +(* Difference *) +Lemma diff_plus_opp : (f1,f2:R->R) (minus_fct f1 f2)==(plus_fct f1 (opp_fct f2)). +Intros; Unfold minus_fct plus_fct opp_fct; Apply fct_eq; Intro x; Ring. +Save. + +Lemma deriv_diff : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> ``(derive_pt (minus_fct f1 f2) x)==(derive_pt f1 x)-(derive_pt f2 x)``. +Intros; Rewrite diff_plus_opp; Unfold Rminus; Rewrite <- (deriv_opposite f2 x H0); Apply deriv_sum; [Assumption | Apply opposite_derivable_pt; Assumption]. +Save. + +Lemma diff_derivable_pt : (f1,f2:R->R;x:R) (derivable_pt f1 x)->(derivable_pt f2 x)->(derivable_pt (minus_fct f1 f2) x). +Intros; Rewrite (diff_plus_opp f1 f2); Apply sum_derivable_pt; [Assumption | Apply opposite_derivable_pt; Assumption]. +Save. + +Lemma diff_derivable : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (minus_fct f1 f2)). +Unfold derivable; Intros f1 f2 H1 H2 x; Apply diff_derivable_pt; [ Exact (H1 x) | Exact (H2 x)]. +Save. + +(**********) +Lemma deriv_scal : (f:R->R;a,x:R) (derivable_pt f x) -> ``(derive_pt (mult_real_fct a f) x)==a*(derive_pt f x)``. +Intros f a x Ha; Unfold mult_real_fct; Generalize (derivable_derive f x Ha); Intro Hb; Elim Hb; Intros l Hc; Rewrite Hc; Apply derive_pt_def_0; Generalize (Req_EM a R0); Intro H0; Elim H0; Intro H1. +Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Rewrite H1; Repeat Rewrite Rmult_Ol; Repeat Rewrite minus_R0; Unfold Rdiv; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Assumption. +Intros; Generalize (derive_pt_def_1 f x l Hc); Intro H2; Elim (H2 ``eps/(Rabsolu a)``). +Intros; Exists x0; Intros; Replace ``(a*(f (x+h))-a*(f x))/h-a*l`` with ``a*(((f (x+h))-(f x))/h-l)``. +Rewrite Rabsolu_mult; Replace ``eps`` with ``(Rabsolu a)*(eps/(Rabsolu a))``. +Apply Rlt_monotony. +Apply (Rabsolu_pos_lt a H1). +Apply (H3 h H4 H5). +Rewrite <- Rmult_sym; Unfold Rdiv; Rewrite Rmult_assoc; Rewrite <- (Rinv_l_sym (Rabsolu a)); [Apply Rmult_1r | Apply (Rabsolu_no_R0 a H1)]. +Field; Assumption. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply (Rabsolu_pos_lt a H1)]. +Save. + +Lemma scal_derivable_pt : (f:R->R;a:R; x:R) (derivable_pt f x) -> +(derivable_pt (mult_real_fct a f) x). +Unfold mult_real_fct derivable_pt; Intros; Generalize (Req_EM a R0); Intro H0; Elim H0; Intro H1. +Intros; Exists ``0``; Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Rewrite H1; Repeat Rewrite Rmult_Ol; Unfold Rminus; Repeat Rewrite Ropp_O; Repeat Rewrite Rplus_Or; Unfold Rdiv; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Assumption. +Elim H; Intros l H2; Exists ``a*l``; Intros; Elim (H2 ``eps/(Rabsolu a)``); Intros. +Exists x0; Intros; Replace ``(a*(f (x+h))-a*(f x))/h-a*l`` with ``a*(((f (x+h))-(f x))/h-l)``. +Rewrite Rabsolu_mult; Replace ``eps`` with ``(Rabsolu a)*(eps/(Rabsolu a))``. +Apply Rlt_monotony. +Apply (Rabsolu_pos_lt a H1). +Apply (H4 h H5 H6). +Rewrite <- Rmult_sym; Unfold Rdiv; Rewrite Rmult_assoc; Rewrite <- (Rinv_l_sym (Rabsolu a)); [Apply Rmult_1r | Apply (Rabsolu_no_R0 a H1)]. +Field; Assumption. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply (Rabsolu_pos_lt a H1)]. +Save. + +Lemma scal_derivable : (f:R->R;a:R) (derivable f) -> (derivable (mult_real_fct a f)). +Unfold derivable; Intros f a H1 x; Apply scal_derivable_pt; Exact +(H1 x). +Save. + +(* Multiplication *) +Lemma deriv_prod : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> ``(derive_pt (mult_fct f1 f2) x)==(derive_pt f1 x)*(f2 x)+(derive_pt f2 x)*(f1 x)``. +Intros; Generalize (derivable_derive f1 x H); Intro; Generalize (derivable_derive f2 x H0); Intro; Elim H1; Clear H1; Intros l1 H1; Elim H2; Clear H2; Intros l2 H2; Cut l1==((fct_cte l1) x). +Cut l2==((fct_cte l2) x). +Intros; Rewrite H3 in H2; Rewrite H4 in H1; Generalize derive_pt_D_in; Intro; Generalize (H5 f1 (fct_cte l1) x); Intro; Generalize (H5 f2 (fct_cte l2) x); Intro; Elim H6; Elim H7; Intros; Generalize (H11 H1); Intro; Generalize (H9 H2); Intro; Rewrite H1; Rewrite H2; Replace ``(fct_cte l1 x)*(f2 x)+(fct_cte l2 x)*(f1 x)`` with ``((plus_fct (mult_fct (fct_cte l1) f2) (mult_fct f1 (fct_cte l2))) x)``. +Generalize (H5 (mult_fct f1 f2) (plus_fct (mult_fct (fct_cte l1) f2) (mult_fct f1 (fct_cte l2))) x); Intro; Elim H14; Intros; Apply H15; Unfold mult_fct plus_fct; Apply Dmult; Assumption. +Unfold plus_fct mult_fct fct_cte; Ring. +Unfold fct_cte; Reflexivity. +Unfold fct_cte; Reflexivity. +Save. + +Lemma prod_derivable_pt : (f1,f2:R->R;x:R) (derivable_pt f1 x)->(derivable_pt f2 x)->(derivable_pt (mult_fct f1 f2) x). +Intros; Generalize (deriv_prod f1 f2 x H H0); Intro; Apply (derive_derivable (mult_fct f1 f2) x ``(derive_pt f1 x)*(f2 x)+(derive_pt f2 x)*(f1 x)`` H1). +Save. + +Lemma prod_derivable : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (mult_fct f1 f2)). +Unfold derivable; Intros f1 f2 H1 H2 x; Apply prod_derivable_pt; [ Exact (H1 x) | Exact (H2 x)]. +Save. + +(**********) +Lemma deriv_const : (a:R;x:R) (derive_pt ([x:R] a) x)==``0``. +Intros; Apply derive_pt_def_0; Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Replace ``a-a`` with ``0``; [Unfold Rdiv; Rewrite Rmult_Ol; Rewrite minus_R0; Rewrite Rabsolu_R0; Assumption | Ring]. +Save. + +Lemma const_derivable : (a:R) (derivable ([x:R] a)). +Unfold derivable; Unfold derivable_pt; Intros; Exists ``0``; Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Unfold Rminus; Rewrite Rplus_Ropp_r; Unfold Rdiv; Rewrite Rmult_Ol; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Save. + +(**********) +Lemma deriv_id : (x:R) (derive_pt ([y:R] y) x)==``1``. +Intro x; Apply derive_pt_def_0; Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Replace ``(x+h-x)/h-1`` with ``0``. +Rewrite Rabsolu_R0; Assumption. +Field; Assumption. +Save. + +Lemma diff_id : (derivable ([x:R] x)). +Unfold derivable; Intro x; Unfold derivable_pt; Exists ``1``; Intros eps Heps; Exists (mkposreal eps Heps); Intros h H1 H2; Replace ``(x+h-x)/h-1`` with ``0``; [Rewrite Rabsolu_R0; Apply Rle_lt_trans with ``(Rabsolu h)``; [Apply Rabsolu_pos | Assumption] | Field; Assumption]. +Save. + +(**********) +Lemma sum_fct_cte_derive_pt : (f:R->R;t,a:R) (derivable_pt f t) -> (derive_pt ([x:R]``(f x)+a``) t)==(derive_pt f t). +Intros; Generalize (derivable_derive f t H); Intro; Elim H0; Intros l H1; Rewrite H1; Apply derive_pt_def_0; Intros; Generalize (derive_pt_def_1 f t l H1); Intros; Elim (H3 eps H2); Intros delta H4; Exists delta; Intros; Replace ``(f (t+h))+a-((f t)+a)`` with ``(f (t+h))-(f t)``; [Apply (H4 h H5 H6) | Ring]. +Save. + +Lemma sum_fct_cte_derivable_pt : (f:R->R;t,a:R) (derivable_pt f t)->(derivable_pt ([t:R]``(f t)+a``) t). +Unfold derivable_pt; Intros; Elim H; Intros; Exists x; Intros; Elim (H0 eps H1); Intros; Exists x0; Intro h; Replace ``(f (t+h))+a-((f t)+a)`` with ``(f (t+h))-(f t)``; [Exact (H2 h) | Ring]. +Save. + +Lemma sum_fct_cte_derivable : (f:R->R;a:R) (derivable f)->(derivable ([t:R]``(f t)+a``)). +Unfold derivable; Intros; Apply sum_fct_cte_derivable_pt; Apply (H x). +Save. + +(**********) +Lemma deriv_Rsqr : (x:R) (derive Rsqr x)==``2*x``. +Intro x; Unfold Rsqr; Unfold derive; Apply (derive_pt_def_0 ([x0:R]``x0*x0``) x); Intros eps Heps; Exists (mkposreal eps Heps); Intros h H1 H2; Replace ``((x+h)*(x+h)-x*x)/h-2*x`` with ``h``; [Assumption | Field; Assumption]. +Save. + +Lemma diff_Rsqr : (derivable Rsqr). +Unfold derivable; Intro x; Unfold Rsqr; Unfold derivable_pt; Exists ``2*x``; Intros eps Heps; Exists (mkposreal eps Heps); Intros h H1 H2; Replace ``((x+h)*(x+h)-x*x)/h-2*x`` with ``h``; [Assumption | Field; Assumption]. +Save. + +Lemma Rsqr_derivable_pt : (f:R->R;t:R) (derivable_pt f t) -> (derivable_pt ([x:R](Rsqr (f x))) t). +Unfold Rsqr; Intros; Generalize (prod_derivable_pt f f t H H); Unfold mult_fct; Intro H0; Assumption. +Save. + +Lemma Rsqr_derivable : (f:R->R) (derivable f)->(derivable ([x:R](Rsqr (f x)))). +Unfold derivable; Intros; Apply (Rsqr_derivable_pt f x (H x)). +Save. + +(* SQRT *) +Axiom deriv_sqrt : (x:R) ``0<x`` -> (derive sqrt)==[y:R] ``1/(2*(sqrt y))``. + +Lemma eq_fct : (x:R;f1,f2:R->R) f1==f2 -> (f1 x)==(f2 x). +Intros; Rewrite H; Reflexivity. +Save. + +Lemma diff_sqrt : (x:R) ``0<x`` -> (derivable_pt sqrt x). +Intros; Generalize (deriv_sqrt x H); Unfold derive; Intro; Generalize (eq_fct x ([x:R](derive_pt sqrt x)) ([y:R]``1/(2*(sqrt y))``) H0); Intro; Apply (derive_derivable sqrt x ``1/(2*(sqrt x))`` H1). +Save. + +(* Composition *) +Axiom prov : (f:R->R) (Dgf no_cond no_cond f)==no_cond. + +Lemma deriv_composition : (f,g:R->R;x:R) (derivable_pt f x) -> (derivable_pt g (f x)) -> ``(derive_pt (comp g f) x)==(derive_pt g (f x))*(derive_pt f x)``. +Intros; Generalize (derivable_derive f x H); Intro; Generalize (derivable_derive g (f x) H0); Intro; Elim H1; Clear H1; Intros l1 H1; Elim H2; Clear H2; Intros l2 H2. +Cut l1==((fct_cte l1) x). +Cut l2==((fct_cte l2) x). +Intros; Rewrite H3 in H2; Rewrite H4 in H1; Rewrite H1; Rewrite H2; Generalize derive_pt_D_in; Intro; Elim (H5 f (fct_cte l1) x); Intros; Elim (H5 g (fct_cte l2) (f x)); Intros; Generalize (H9 H2); Intro; Generalize (H7 H1); Intro; Replace ``(fct_cte l2 x)*(fct_cte l1 x)`` with ``((mult_fct (fct_cte l1) (fct_cte l2)) x)``. +Elim (H5 (comp g f) (mult_fct (fct_cte l1) (fct_cte l2)) x); Intros; Apply H12. +Unfold comp mult_fct; Generalize (prov f); Intro; Rewrite <- H14; Apply (Dcomp no_cond no_cond (fct_cte l1) (fct_cte l2) f g x); Assumption. +Unfold mult_fct fct_cte; Rewrite Rmult_sym; Reflexivity. +Unfold fct_cte; Reflexivity. +Unfold fct_cte; Reflexivity. +Save. + +Lemma composition_derivable : (f,g:R->R;x:R) (derivable_pt f x) -> (derivable_pt g (f x)) -> (derivable_pt (comp g f) x). +Intros; Generalize (deriv_composition f g x H H0); Intro; Apply (derive_derivable (comp g f) x ``(derive_pt g (f x))*(derive_pt f x)`` H1). +Save. + +Lemma derive_composition : (f,g:R->R;x:R) (derivable_pt f x) -> (derivable_pt g (f x)) -> (derive_pt ([x:R]``(g (f x))``) x)==``(derive_pt g (f x))*(derive_pt f x)``. +Intros; Generalize (deriv_composition f g x H H0); Unfold comp; Intro; Assumption. +Save. + +Lemma composition_derivable_var : (f,g:R->R;x:R) (derivable_pt f x) -> (derivable_pt g (f x)) -> (derivable_pt ([x:R](g (f x))) x). +Intros; Generalize (composition_derivable f g x H H0); Unfold comp; Intro; Assumption. +Save. + +Lemma diff_comp : (f,g:R->R) (derivable f)->(derivable g)->(derivable (comp g f)). +Intros f g; Unfold derivable; Intros H1 H2 x; Apply (composition_derivable f g x (H1 x) (H2 (f x))). +Save. + +Lemma Rsqr_derive : (f:R->R;t:R) (derivable_pt f t)->(derive_pt ([x:R](Rsqr (f x))) t)==(Rmult ``2`` (Rmult (derive_pt f t) (f t))). +Intros; Generalize diff_Rsqr; Unfold derivable; Intro H0; Generalize (deriv_composition f Rsqr t H (H0 (f t))); Unfold comp; Intro H1; Rewrite H1; Generalize (deriv_Rsqr (f t)); Unfold derive; Intro H2; Rewrite H2; Rewrite Rmult_assoc; Rewrite <- (Rmult_sym (derive_pt f t)); Reflexivity. +Save. + +(* SIN and COS *) +Axiom deriv_sin : (derive sin)==cos. + +Lemma diff_sin : (derivable sin). +Unfold derivable; Intro; Generalize deriv_sin; Unfold derive; Intro; Generalize +(eq_fct x ([x:R](derive_pt sin x)) cos H); Intro; Apply (derive_derivable sin x +(cos x) H0). +Save. + +Lemma diff_cos : (derivable cos). +Unfold derivable; Intro; Cut ([x:R]``(sin (x+PI/2))``)==cos. +Intro; Rewrite <- H; Apply (composition_derivable_var ([x:R]``x+PI/2``) sin x). +Apply (sum_fct_cte_derivable_pt ([x:R]x) x ``PI/2``); Apply diff_id. +Apply diff_sin. +Apply fct_eq; Intro; Symmetry; Rewrite Rplus_sym; Apply cos_sin. +Save. + +Lemma derive_pt_sin : (x:R) (derive_pt sin x)==(cos x). +Intro; Generalize deriv_sin; Unfold derive; Intro; Apply (eq_fct x [x:R](derive_pt sin x) cos H). +Save. + +Lemma deriv_cos : (derive cos)==(opp_fct sin). +Unfold opp_fct derive; Apply fct_eq; Intro; Cut ([x:R]``(sin (x+PI/2))``)==cos. +Intro; Rewrite <- H; Rewrite (derive_composition ([x:R]``x+PI/2``) sin x). +Rewrite (derive_pt_sin ``x+PI/2``); Rewrite (sum_fct_cte_derive_pt ([x:R]``x``) x ``PI/2``). +Generalize (deriv_id x); Intro; Unfold derive in H0; Rewrite H0; Rewrite Rmult_1r; Rewrite Rplus_sym; Rewrite sin_cos; Rewrite Ropp_Ropp; Reflexivity. +Apply diff_id. +Apply (sum_fct_cte_derivable_pt ([x:R]x) x ``PI/2``); Apply diff_id. +Apply diff_sin. +Apply fct_eq; Intro; Symmetry; Rewrite Rplus_sym; Apply cos_sin. +Save. + +Lemma derive_pt_cos : (x:R) (derive_pt cos x)==``-(sin x)``. +Intro; Generalize deriv_cos; Unfold derive; Intro; Unfold opp_fct in H; Apply (eq_fct x [x:R](derive_pt cos x) [x:R]``-(sin x)`` H). +Save. + +(************************************************************) +(* Local extremum's condition *) +(************************************************************) +Theorem deriv_maximum : (f:R->R;a,b,c:R) ``a<c``->``c<b``->(derivable_pt f c)->((x:R) ``a<x``->``x<b``->``(f x)<=(f c)``)->``(derive_pt f c)==0``. +Intros; Case (total_order R0 (derive_pt f c)); Intro. +Generalize (derivable_derive f c H1); Intro; Elim H4; Intros l H5; Rewrite H5 in H3; Generalize (derive_pt_def_1 f c l H5); Intro. +Cut ``0<l/2``. +Intro; Elim (H6 ``l/2`` H7); Intros delta H8. +Cut ``0<(b-c)/2``. +Intro; Cut ``(Rmin delta/2 ((b-c)/2))<>0``. +Intro; Cut ``(Rabsolu (Rmin delta/2 ((b-c)/2)))<delta``. +Intro; Generalize (H8 ``(Rmin delta/2 ((b-c)/2))`` H10 H11); Intro; Cut ``0<(Rmin (delta/2) ((b-c)/2))``. +Intro; Cut ``a<c+(Rmin (delta/2) ((b-c)/2))``. +Cut ``c+(Rmin (delta/2) ((b-c)/2))<b``. +Intros; Generalize (H2 ``c+(Rmin (delta/2) ((b-c)/2))`` H15 H14); Intro; Cut ``((f (c+(Rmin (delta/2) ((b-c)/2))))-(f c))/(Rmin (delta/2) ((b-c)/2))<=0``. +Intro; Cut ``-l<0``. +Intro; Unfold Rminus in H12. +Cut ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l<0``. +Intro; Cut ``(Rabsolu (((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l)) < l/2``. +Unfold Rabsolu; Case (case_Rabsolu ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l``); Intro. +Replace `` -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l)`` with ``l+ -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2)))``. +Intro; Generalize (Rlt_compatibility ``-l`` ``l+ -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2)))`` ``l/2`` H20); Repeat Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Replace ``-l+l/2`` with ``-(l/2)``. +Intro; Generalize (Rlt_Ropp ``-(((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2)))`` ``-(l/2)`` H21); Repeat Rewrite Ropp_Ropp; Intro; Generalize (Rlt_trans ``0`` ``l/2`` ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))`` H7 H22); Intro; Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))`` ``0`` H23 H17)). +Field; DiscrR. +Ring. +Intro; Generalize (Rle_sym2 ``0`` ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l`` r); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l`` ``0`` H21 H19)). +Assumption. +Rewrite <- Ropp_O; Replace ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l`` with ``-(l+ -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))-(f c))/(Rmin (delta/2) ((b+ -c)/2))))``. +Apply Rgt_Ropp; Change ``0<l+ -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))-(f c))/(Rmin (delta/2) ((b+ -c)/2)))``; Apply gt0_plus_ge0_is_gt0; [Assumption | Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Assumption]. +Ring. +Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. +Replace ``((f (c+(Rmin (delta/2) ((b-c)/2))))-(f c))/(Rmin (delta/2) ((b-c)/2))`` with ``- (((f c)-(f (c+(Rmin (delta/2) ((b-c)/2)))))/(Rmin (delta/2) ((b-c)/2)))``. +Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Unfold Rdiv; Apply Rmult_le_pos; [Generalize (Rle_compatibility_r ``-(f (c+(Rmin (delta*/2) ((b-c)*/2))))`` ``(f (c+(Rmin (delta*/2) ((b-c)*/2))))`` (f c) H16); Rewrite Rplus_Ropp_r; Intro; Assumption | Left; Apply Rlt_Rinv; Assumption]. +Field; Assumption. +Generalize (Rmin_r ``(delta/2)`` ``((b-c)/2)``); Intro; Generalize (Rle_compatibility ``c`` ``(Rmin (delta/2) ((b-c)/2))`` ``(b-c)/2`` H14); Intro; Apply Rle_lt_trans with ``c+(b-c)/2``. +Assumption. +Apply Rlt_monotony_contra with ``2``. +Apply Rgt_2_0. +Replace ``2*(c+(b-c)/2)`` with ``c+b``. +Replace ``2*b`` with ``b+b``. +Apply Rlt_compatibility_r; Assumption. +Ring. +Field; DiscrR. +Apply Rlt_trans with c. +Assumption. +Pattern 1 c; Rewrite <- (Rplus_Or c); Apply Rlt_compatibility; Assumption. +Cut ``0<delta/2``. +Intro; Apply (Rmin_stable_in_posreal (mkposreal ``delta/2`` H13) (mkposreal ``(b-c)/2`` H9)). +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. +Unfold Rabsolu; Case (case_Rabsolu (Rmin ``delta/2`` ``(b-c)/2``)). +Intro. +Cut ``0<delta/2``. +Intro. +Generalize (Rmin_stable_in_posreal (mkposreal ``delta/2`` H11) (mkposreal ``(b-c)/2`` H9)); Simpl; Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``(Rmin (delta/2) ((b-c)/2))`` ``0`` H12 r)). +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. +Intro; Apply Rle_lt_trans with ``delta/2``. +Apply Rmin_l. +Unfold Rdiv; Apply Rlt_monotony_contra with ``2``. +Apply Rgt_2_0. +Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l. +Replace ``2*delta`` with ``delta+delta``. +Pattern 2 delta; Rewrite <- (Rplus_Or delta); Apply Rlt_compatibility. +Rewrite Rplus_Or; Apply (cond_pos delta). +Ring. +DiscrR. +Cut ``0<delta/2``. +Intro; Generalize (Rmin_stable_in_posreal (mkposreal ``delta/2`` H10) (mkposreal ``(b-c)/2`` H9)); Simpl; Intro; Red; Intro; Rewrite H12 in H11; Elim (Rlt_antirefl ``0`` H11). +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. +Unfold Rdiv; Apply Rmult_lt_pos. +Generalize (Rlt_compatibility_r ``-c`` c b H0); Rewrite Rplus_Ropp_r; Intro; Assumption. +Apply Rlt_Rinv; Apply Rgt_2_0. +Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_2_0]. +Elim H3; Intro. +Symmetry; Assumption. +Generalize (derivable_derive f c H1); Intro; Elim H5; Intros l H6; Rewrite H6 in H4; Generalize (derive_pt_def_1 f c l H6); Intro; Cut ``0< -(l/2)``. +Intro; Elim (H7 ``-(l/2)`` H8); Intros delta H9. +Cut ``0<(c-a)/2``. +Intro; Cut ``(Rmax (-(delta/2)) ((a-c)/2))<0``. +Intro; Cut ``(Rmax (-(delta/2)) ((a-c)/2))<>0``. +Intro; Cut ``(Rabsolu (Rmax (-(delta/2)) ((a-c)/2)))<delta``. +Intro; Generalize (H9 ``(Rmax (-(delta/2)) ((a-c)/2))`` H12 H13); Intro; Cut ``a<c+(Rmax (-(delta/2)) ((a-c)/2))``. +Cut ``c+(Rmax (-(delta/2)) ((a-c)/2))<b``. +Intros; Generalize (H2 ``c+(Rmax (-(delta/2)) ((a-c)/2))`` H16 H15); Intro; Cut ``0<=((f (c+(Rmax (-(delta/2)) ((a-c)/2))))-(f c))/(Rmax (-(delta/2)) ((a-c)/2))``. +Intro; Cut ``0< -l``. +Intro; Unfold Rminus in H14; Cut ``0<((f (c+(Rmax (-(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax (-(delta/2)) ((a+ -c)/2))+ -l``. +Intro; Cut ``(Rabsolu (((f (c+(Rmax (-(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax (-(delta/2)) ((a+ -c)/2))+ -l)) < -(l/2)``. +Unfold Rabsolu; Case (case_Rabsolu ``((f (c+(Rmax (-(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax (-(delta/2)) ((a+ -c)/2))+ -l``). +Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``((f (c+(Rmax ( -(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax ( -(delta/2)) ((a+ -c)/2))+ -l`` ``0`` H20 r)). +Intros; Generalize (Rlt_compatibility_r ``l`` ``(((f (c+(Rmax (-(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax (-(delta/2)) ((a+ -c)/2)))+ -l`` ``-(l/2)`` H21); Repeat Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Replace ``-(l/2)+l`` with ``l/2``. +Cut ``l/2<0``. +Intros; Generalize (Rlt_trans ``((f (c+(Rmax ( -(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax ( -(delta/2)) ((a+ -c)/2))`` ``l/2`` ``0`` H23 H22); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``((f (c+(Rmax ( -(delta/2)) ((a-c)/2))))-(f c))/(Rmax ( -(delta/2)) ((a-c)/2))`` ``0`` H18 H24)). +Rewrite <- (Ropp_Ropp ``l/2``); Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. +Field; DiscrR. +Assumption. +Apply ge0_plus_gt0_is_gt0; Assumption. +Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. +Unfold Rdiv; Replace ``((f (c+(Rmax ( -(delta*/2)) ((a-c)*/2))))-(f c))*/(Rmax ( -(delta*/2)) ((a-c)*/2))`` with ``(-((f (c+(Rmax ( -(delta*/2)) ((a-c)*/2))))-(f c)))*/(-(Rmax ( -(delta*/2)) ((a-c)*/2)))``. +Apply Rmult_le_pos. +Generalize (Rle_compatibility ``-(f (c+(Rmax (-(delta*/2)) ((a-c)*/2))))`` ``(f (c+(Rmax (-(delta*/2)) ((a-c)*/2))))`` (f c) H17); Rewrite Rplus_Ropp_l; Replace ``-((f (c+(Rmax ( -(delta*/2)) ((a-c)*/2))))-(f c))`` with ``-((f (c+(Rmax ( -(delta*/2)) ((a-c)*/2)))))+(f c)``. +Intro; Assumption. +Ring. +Left; Apply Rlt_Rinv; Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. +Field; Apply prod_neq_R0; [Apply Ropp_neq; Assumption | Assumption]. +Generalize (Rlt_compatibility c ``(Rmax ( -(delta/2)) ((a-c)/2))`` ``0`` H11); Rewrite Rplus_Or; Intro; Apply Rlt_trans with ``c``; Assumption. +Generalize (RmaxLess2 ``(-(delta/2))`` ``((a-c)/2)``); Intro; Generalize (Rle_compatibility c ``(a-c)/2`` ``(Rmax ( -(delta/2)) ((a-c)/2))`` H15); Intro; Apply Rlt_le_trans with ``c+(a-c)/2``. +Apply Rlt_monotony_contra with ``2``. +Apply Rgt_2_0. +Replace ``2*(c+(a-c)/2)`` with ``a+c``. +Replace ``2*a`` with ``a+a``. +Apply Rlt_compatibility; Assumption. +Ring. +Field; DiscrR. +Assumption. +Unfold Rabsolu; Case (case_Rabsolu (Rmax ``-(delta/2)`` ``(a-c)/2``)). +Intro; Generalize (RmaxLess1 ``-(delta/2)`` ``(a-c)/2``); Intro; Generalize (Rle_Ropp ``-(delta/2)`` ``(Rmax ( -(delta/2)) ((a-c)/2))`` H13); Rewrite Ropp_Ropp; Intro; Generalize (Rle_sym2 ``-(Rmax ( -(delta/2)) ((a-c)/2))`` ``delta/2`` H14); Intro; Apply Rle_lt_trans with ``delta/2``. +Assumption. +Apply Rlt_monotony_contra with ``2``. +Apply Rgt_2_0. +Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l; Replace ``2*delta`` with ``delta+delta``. +Pattern 2 delta; Rewrite <- (Rplus_Or delta); Apply Rlt_compatibility; Rewrite Rplus_Or; Apply (cond_pos delta). +Ring. +DiscrR. +Cut ``-(delta/2) < 0``. +Cut ``(a-c)/2<0``. +Intros; Generalize (Rmax_stable_in_negreal (mknegreal ``-(delta/2)`` H14) (mknegreal ``(a-c)/2`` H13)); Simpl; Intro; Generalize (Rle_sym2 ``0`` ``(Rmax ( -(delta/2)) ((a-c)/2))`` r); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``(Rmax ( -(delta/2)) ((a-c)/2))`` ``0`` H16 H15)). +Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp ``(a-c)/2``); Apply Rlt_Ropp; Replace ``-((a-c)/2)`` with ``(c-a)/2``; [Assumption | Field; DiscrR]. +Rewrite <- Ropp_O; Apply Rlt_Ropp; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply (Rlt_Rinv ``2`` Rgt_2_0)]. +Red; Intro; Rewrite H12 in H11; Elim (Rlt_antirefl ``0`` H11). +Cut ``(a-c)/2<0``. +Intro; Cut ``-(delta/2)<0``. +Intro; Apply (Rmax_stable_in_negreal (mknegreal ``-(delta/2)`` H12) (mknegreal ``(a-c)/2`` H11)). +Rewrite <- Ropp_O; Apply Rlt_Ropp; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply (Rlt_Rinv ``2`` Rgt_2_0)]. +Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp ``(a-c)/2``); Apply Rlt_Ropp; Replace ``-((a-c)/2)`` with ``(c-a)/2``; [Assumption | Field; DiscrR]. +Unfold Rdiv; Apply Rmult_lt_pos; [Generalize (Rlt_compatibility_r ``-a`` a c H); Rewrite Rplus_Ropp_r; Intro; Assumption | Apply (Rlt_Rinv ``2`` Rgt_2_0)]. +Replace ``-(l/2)`` with ``(-l)/2``; [Unfold Rdiv; Apply Rmult_lt_pos; [Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption | Apply (Rlt_Rinv ``2`` Rgt_2_0)] | Field; DiscrR]. +Save. + +Theorem deriv_minimum : (f:R->R;a,b,c:R) ``a<c``->``c<b``->(derivable_pt f c)->((x:R) ``a<x``->``x<b``->``(f c)<=(f x)``)->``(derive_pt f c)==0``. +Intros; Generalize (opposite_derivable_pt f c H1); Intro; Rewrite <- (Ropp_Ropp (derive_pt f c)); Apply eq_RoppO; Rewrite <- (deriv_opposite f c H1); Apply (deriv_maximum (opp_fct f) a b c H H0 H3); Intros; Unfold opp_fct; Apply Rge_Ropp; Apply Rle_sym1; Apply (H2 x H4 H5). +Save. + +Theorem deriv_constant2 : (f:R->R;a,b,c:R) ``a<c``->``c<b``->(derivable_pt f c)->((x:R) ``a<x``->``x<b``->``(f x)==(f c)``)->``(derive_pt f c)==0``. +Intros; Apply (deriv_maximum f a b c H H0 H1); Intros; Right; Apply (H2 x H3 H4). +Save. + +(**********) +Lemma nonneg_derivative_0 : (f:R->R) (derivable f)->(increasing f) -> ((x:R) ``0<=(derive_pt f x)``). +Intros; Unfold increasing in H0; Generalize (derivable_derive f x (H x)); Intro; Elim H1; Intros l H2. +Rewrite H2; Case (total_order R0 l); Intro. +Left; Assumption. +Elim H3; Intro. +Right; Assumption. +Generalize (derive_pt_def_1 f x l H2); Intros; Cut ``0< -(l/2)``. +Intro; Elim (H5 ``-(l/2)`` H6); Intros delta H7; Cut ``delta/2<>0``/\``0<delta/2``/\``(Rabsolu delta/2)<delta``. +Intro; Decompose [and] H8; Intros; Generalize (H7 ``delta/2`` H9 H12); Cut ``0<=((f (x+delta/2))-(f x))/(delta/2)``. +Intro; Cut ``0<=((f (x+delta/2))-(f x))/(delta/2)-l``. +Intro; Unfold Rabsolu; Case (case_Rabsolu ``((f (x+delta/2))-(f x))/(delta/2)-l``). +Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``((f (x+delta/2))-(f x))/(delta/2)-l`` ``0`` H13 r)). +Intros; Generalize (Rlt_compatibility_r l ``((f (x+delta/2))-(f x))/(delta/2)-l`` ``-(l/2)`` H14); Unfold Rminus; Replace ``-(l/2)+l`` with ``l/2``. +Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Intro; Generalize (Rle_lt_trans ``0`` ``((f (x+delta/2))-(f x))/(delta/2)`` ``l/2`` H10 H15); Intro; Cut ``l/2<0``. +Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``l/2`` ``0`` H16 H17)). +Rewrite <- Ropp_O in H6; Generalize (Rlt_Ropp ``-0`` ``-(l/2)`` H6); Repeat Rewrite Ropp_Ropp; Intro; Assumption. +Field; DiscrR. +Unfold Rminus; Apply ge0_plus_ge0_is_ge0. +Unfold Rdiv; Apply Rmult_le_pos. +Cut ``x<=(x+(delta*/2))``. +Intro; Generalize (H0 x ``x+(delta*/2)`` H13); Intro; Generalize (Rle_compatibility ``-(f x)`` ``(f x)`` ``(f (x+delta*/2))`` H14); Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Intro; Assumption. +Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Left; Assumption. +Left; Apply Rlt_Rinv; Assumption. +Left; Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. +Unfold Rdiv; Apply Rmult_le_pos. +Cut ``x<=(x+(delta*/2))``. +Intro; Generalize (H0 x ``x+(delta*/2)`` H10); Intro; Generalize (Rle_compatibility ``-(f x)`` ``(f x)`` ``(f (x+delta*/2))`` H13); Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Intro; Assumption. +Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Left; Assumption. +Left; Apply Rlt_Rinv; Assumption. +Split. +Unfold Rdiv; Apply prod_neq_R0. +Generalize (cond_pos delta); Intro; Red; Intro H9; Rewrite H9 in H8; Elim (Rlt_antirefl ``0`` H8). +Apply Rinv_neq_R0; DiscrR. +Split. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. +Unfold Rabsolu; Case (case_Rabsolu ``delta/2``). +Unfold Rdiv; Intro; Generalize (Rlt_monotony_r ``2`` ``delta*/2`` ``0`` Rgt_2_0 r); Rewrite Rmult_Ol; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` delta ``0`` (cond_pos delta) H8)). +DiscrR. +Intro; Unfold Rdiv; Pattern 1 delta; Replace ``(pos delta)`` with ``2*(delta*/2)``. +Replace ``2*(delta*/2)`` with ``delta*/2+delta*/2``. +Pattern 2 delta; Rewrite <- (Rplus_Or ``delta*/2``). +Apply Rlt_compatibility. +Rewrite Rplus_Or. +Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. +Ring. +Field; DiscrR. +Rewrite <- Ropp_O; Apply Rlt_Ropp; Unfold Rdiv; Generalize (Rlt_monotony_r ``/2`` l ``0`` (Rlt_Rinv ``2`` Rgt_2_0) H4); Rewrite Rmult_Ol; Intro; Assumption. +Save. + +(**********) +Axiom nonneg_derivative_1 : (f:R->R) (derivable f)->((x:R) ``0<=(derive_pt f x)``) -> (increasing f). + +(**********) +Lemma nonpos_derivative_0 : (f:R->R) (derivable f)->(decreasing f) -> ((x:R) ``(derive_pt f x)<=0``). +Intros; Unfold decreasing in H0; Generalize (derivable_derive f x (H x)); Intro; Elim H1; Intros l H2. +Rewrite H2; Case (total_order l R0); Intro. +Left; Assumption. +Elim H3; Intro. +Right; Assumption. +Generalize (derive_pt_def_1 f x l H2); Intros; Cut ``0< (l/2)``. +Intro; Elim (H5 ``(l/2)`` H6); Intros delta H7; Cut ``delta/2<>0``/\``0<delta/2``/\``(Rabsolu delta/2)<delta``. +Intro; Decompose [and] H8; Intros; Generalize (H7 ``delta/2`` H9 H12); Cut ``((f (x+delta/2))-(f x))/(delta/2)<=0``. +Intro; Cut ``0< -(((f (x+delta/2))-(f x))/(delta/2)-l)``. +Intro; Unfold Rabsolu; Case (case_Rabsolu ``((f (x+delta/2))-(f x))/(delta/2)-l``). +Intros; Generalize (Rlt_compatibility_r ``-l`` ``-(((f (x+delta/2))-(f x))/(delta/2)-l)`` ``(l/2)`` H14); Unfold Rminus. +Replace ``(l/2)+ -l`` with ``-(l/2)``. +Replace `` -(((f (x+delta/2))+ -(f x))/(delta/2)+ -l)+ -l`` with ``-(((f (x+delta/2))+ -(f x))/(delta/2))``. +Intro. +Generalize (Rlt_Ropp ``-(((f (x+delta/2))+ -(f x))/(delta/2))`` ``-(l/2)`` H15). +Repeat Rewrite Ropp_Ropp. +Intro. +Generalize (Rlt_trans ``0`` ``l/2`` ``((f (x+delta/2))-(f x))/(delta/2)`` H6 H16); Intro. +Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``((f (x+delta/2))-(f x))/(delta/2)`` ``0`` H17 H10)). +Ring. +Field; DiscrR. +Intros. +Generalize (Rge_Ropp ``((f (x+delta/2))-(f x))/(delta/2)-l`` ``0`` r). +Rewrite Ropp_O. +Intro. +Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``-(((f (x+delta/2))-(f x))/(delta/2)-l)`` ``0`` H13 H15)). +Replace ``-(((f (x+delta/2))-(f x))/(delta/2)-l)`` with ``(((f (x))-(f (x+delta/2)))/(delta/2)) +l``. +Unfold Rminus. +Apply ge0_plus_gt0_is_gt0. +Unfold Rdiv; Apply Rmult_le_pos. +Cut ``x<=(x+(delta*/2))``. +Intro; Generalize (H0 x ``x+(delta*/2)`` H13); Intro; Generalize (Rle_compatibility ``-(f (x+delta/2))`` ``(f (x+delta/2))`` ``(f x)`` H14); Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Intro; Assumption. +Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Left; Assumption. +Left; Apply Rlt_Rinv; Assumption. +Assumption. +Field. +DiscrR. +Case delta; Intros. +Apply prod_neq_R0. +Red; Intro H13; Rewrite H13 in cond_pos; Elim (Rlt_antirefl ``0`` cond_pos). +Apply Rinv_neq_R0; DiscrR. +Replace ``((f (x+delta/2))-(f x))/(delta/2)`` with ``-(((f x)-(f (x+delta/2)))/(delta/2))``. +Rewrite <- Ropp_O. +Apply Rge_Ropp. +Apply Rle_sym1. +Unfold Rdiv; Apply Rmult_le_pos. +Cut ``x<=(x+(delta*/2))``. +Intro; Generalize (H0 x ``x+(delta*/2)`` H10); Intro. +Generalize (Rle_compatibility ``-(f (x+delta/2))`` ``(f (x+delta/2))`` ``(f x)`` H13); Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Intro; Assumption. +Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Left; Assumption. +Left; Apply Rlt_Rinv; Assumption. +Field. +Case delta; Intros. +Apply prod_neq_R0. +Red; Intro H13; Rewrite H13 in cond_pos; Elim (Rlt_antirefl ``0`` cond_pos). +Apply Rinv_neq_R0; DiscrR. +Split. +Unfold Rdiv; Apply prod_neq_R0. +Generalize (cond_pos delta); Intro; Red; Intro H9; Rewrite H9 in H8; Elim (Rlt_antirefl ``0`` H8). +Apply Rinv_neq_R0; DiscrR. +Split. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. +Unfold Rabsolu; Case (case_Rabsolu ``delta/2``). +Unfold Rdiv; Intro; Generalize (Rlt_monotony_r ``2`` ``delta*/2`` ``0`` Rgt_2_0 r); Rewrite Rmult_Ol; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` delta ``0`` (cond_pos delta) H8)). +DiscrR. +Intro; Unfold Rdiv; Pattern 1 delta; Replace ``(pos delta)`` with ``2*(delta*/2)``. +Replace ``2*(delta*/2)`` with ``delta*/2+delta*/2``. +Pattern 2 delta; Rewrite <- (Rplus_Or ``delta*/2``). +Apply Rlt_compatibility. +Rewrite Rplus_Or. +Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. +Ring. +Field; DiscrR. +Unfold Rdiv; Apply Rmult_lt_pos. +Assumption. +Apply Rlt_Rinv; Apply Rgt_2_0. +Save. + +(**********) +Lemma increasing_decreasing_opp : (f:R->R) (increasing f) -> (decreasing (opp_fct f)). +Unfold increasing decreasing opp_fct; Intros; Generalize (H x y H0); Intro; Apply Rge_Ropp; Apply Rle_sym1; Assumption. +Save. + +(**********) +Lemma opp_opp_fct : (f:R->R) (opp_fct (opp_fct f))==f. +Intro; Unfold opp_fct; Apply fct_eq; Intro; Rewrite Ropp_Ropp; Reflexivity. +Save. + +(**********) +Lemma nonpos_derivative_1 : (f:R->R) (derivable f)->((x:R) ``(derive_pt f x)<=0``) -> (decreasing f). +Intros; Rewrite <- (opp_opp_fct f); Apply increasing_decreasing_opp. +Cut (derivable (opp_fct f)). +Cut (x:R)``0<=(derive_pt (opp_fct f) x)``. +Intros; Apply (nonneg_derivative_1 (opp_fct f) H2 H1). +Intros; Rewrite (deriv_opposite f x (H x)); Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Apply (H0 x). +Apply (opposite_derivable f H). +Save. + +(**********) +Axiom positive_derivative : (f:R->R) (derivable f)->((x:R) ``0<(derive_pt f x)``)->(strict_increasing f). + +(**********) +Lemma strictincreasing_strictdecreasing_opp : (f:R->R) (strict_increasing f) -> (strict_decreasing (opp_fct f)). +Unfold strict_increasing strict_decreasing opp_fct; Intros; Generalize (H x y H0); Intro; Apply Rlt_Ropp; Assumption. +Save. + +(**********) +Lemma negative_derivative : (f:R->R) (derivable f)->((x:R) ``(derive_pt f x)<0``)->(strict_decreasing f). +Intros; Rewrite <- (opp_opp_fct f); Apply strictincreasing_strictdecreasing_opp. +Cut (derivable (opp_fct f)). +Cut (x:R)``0<(derive_pt (opp_fct f) x)``. +Intros; Apply (positive_derivative (opp_fct f) H2 H1). +Intros; Rewrite (deriv_opposite f x (H x)); Rewrite <- Ropp_O; Apply Rlt_Ropp; Apply (H0 x). +Apply (opposite_derivable f H). +Save. + +(**********) +Lemma null_derivative_0 : (f:R->R) (constant f)->((x:R) ``(derive_pt f x)==0``). +Intros; Unfold constant in H; Apply derive_pt_def_0; Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Rewrite (H x ``x+h``); Unfold Rminus; Unfold Rdiv; Rewrite Rplus_Ropp_r; Rewrite Rmult_Ol; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. +Save. + +(**********) +Lemma increasing_decreasing : (f:R->R) (increasing f) -> (decreasing f) -> (constant f). +Unfold increasing decreasing constant; Intros; Case (total_order x y); Intro. +Generalize (Rlt_le x y H1); Intro; Apply (Rle_antisym (f x) (f y) (H x y H2) (H0 x y H2)). +Elim H1; Intro. +Rewrite H2; Reflexivity. +Generalize (Rlt_le y x H2); Intro; Symmetry; Apply (Rle_antisym (f y) (f x) (H y x H3) (H0 y x H3)). +Save. + +(**********) +Lemma null_derivative_1 : (f:R->R) (derivable f)->((x:R) ``(derive_pt f x)==0``)->(constant f). +Intros. +Cut (x:R)``(derive_pt f x) <= 0``. +Cut (x:R)``0 <= (derive_pt f x)``. +Intros. +Generalize (nonneg_derivative_1 f H H1); Intro. +Generalize (nonpos_derivative_1 f H H2); Intro. +Apply increasing_decreasing; Assumption. +Intro. +Right; Symmetry; Apply (H0 x). +Intro; Right; Apply (H0 x). +Save. + +(**********) +(**********) +Axiom IAF : (f,g:R->R;a,b:R) ``a<=b`` -> (derivable f) -> (derivable g) -> ((c:R) ``a<=c<=b`` -> ``(derive_pt g c)<=(derive_pt f c)``) -> ``(g b)-(g a)<=(f b)-(f a)``. |