(************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* R. (****************************************************) (** 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))``. Definition inv_fct [f:R->R] : R->R := [x:R]``/(f x)``. V8Infix "+" plus_fct : Rfun_scope. V8Notation "- x" := (opp_fct x) : Rfun_scope. V8Infix "*" mult_fct : Rfun_scope. V8Infix "-" minus_fct : Rfun_scope. V8Infix "/" div_fct : Rfun_scope. Notation Local "f1 'o' f2" := (comp f1 f2) (at level 2, right associativity) : Rfun_scope V8only (at level 20, right associativity). V8Notation "/ x" := (inv_fct x) : Rfun_scope. Delimits Scope Rfun_scope with F. Definition fct_cte [a:R] : R->R := [x:R]a. Definition id := [x:R]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``(f x)<(f y)``. Definition strict_decreasing [f:R->R] : Prop := (x,y:R) ``x``(f y)<(f x)``. Definition constant [f:R->R] : Prop := (x,y:R) ``(f x)==(f y)``. (**********) Definition no_cond : R->Prop := [x:R] True. (**********) Definition constant_D_eq [f:R->R;D:R->Prop;c:R] : Prop := (x:R) (D x) -> (f x)==c. (***************************************************) (** Definition of continuity as a limit *) (***************************************************) (**********) Definition continuity_pt [f:R->R; x0:R] : Prop := (continue_in f no_cond x0). Definition continuity [f:R->R] : Prop := (x:R) (continuity_pt f x). Arguments Scope continuity_pt [Rfun_scope R_scope]. Arguments Scope continuity [Rfun_scope]. (**********) Lemma continuity_pt_plus : (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. Qed. Lemma continuity_pt_opp : (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. Qed. Lemma continuity_pt_minus : (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. Qed. Lemma continuity_pt_mult : (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. Qed. Lemma continuity_pt_const : (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]. Qed. Lemma continuity_pt_scal : (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. Qed. Lemma continuity_pt_inv : (f:R->R; x0:R) (continuity_pt f x0) -> ~``(f x0)==0`` -> (continuity_pt (inv_fct f) x0). Intros. Replace (inv_fct f) with [x:R]``/(f x)``. Unfold continuity_pt; Unfold continue_in; Intros; Apply limit_inv; Assumption. Unfold inv_fct; Reflexivity. Qed. Lemma div_eq_inv : (f1,f2:R->R) (div_fct f1 f2)==(mult_fct f1 (inv_fct f2)). Intros; Reflexivity. Qed. Lemma continuity_pt_div : (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 continuity_pt_mult; [Assumption | Apply continuity_pt_inv; Assumption]. Qed. Lemma continuity_pt_comp : (f1,f2:R->R;x:R) (continuity_pt f1 x) -> (continuity_pt f2 (f1 x)) -> (continuity_pt (comp f2 f1) x). Unfold continuity_pt; Unfold continue_in; Intros; Unfold comp. Cut (limit1_in [x0:R](f2 (f1 x0)) (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1) (f2 (f1 x)) x) -> (limit1_in [x0:R](f2 (f1 x0)) (D_x no_cond x) (f2 (f1 x)) x). Intro; Apply H1. EApply limit_comp. Apply H. Apply H0. Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. Assert H3 := (H1 eps H2). Elim H3; Intros. Exists x0. Split. Elim H4; Intros; Assumption. Intros; Case (Req_EM (f1 x) (f1 x1)); Intro. Rewrite H6; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. Elim H4; Intros; Apply H8. Split. Unfold Dgf D_x no_cond. Split. Split. Trivial. Elim H5; Unfold D_x no_cond; Intros. Elim H9; Intros; Assumption. Split. Trivial. Assumption. Elim H5; Intros; Assumption. Qed. (**********) Lemma continuity_plus : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (plus_fct f1 f2)). Unfold continuity; Intros; Apply (continuity_pt_plus f1 f2 x (H x) (H0 x)). Qed. Lemma continuity_opp : (f:R->R) (continuity f)->(continuity (opp_fct f)). Unfold continuity; Intros; Apply (continuity_pt_opp f x (H x)). Qed. Lemma continuity_minus : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (minus_fct f1 f2)). Unfold continuity; Intros; Apply (continuity_pt_minus f1 f2 x (H x) (H0 x)). Qed. Lemma continuity_mult : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (mult_fct f1 f2)). Unfold continuity; Intros; Apply (continuity_pt_mult f1 f2 x (H x) (H0 x)). Qed. Lemma continuity_const : (f:R->R) (constant f) -> (continuity f). Unfold continuity; Intros; Apply (continuity_pt_const f x H). Qed. Lemma continuity_scal : (f:R->R;a:R) (continuity f) -> (continuity (mult_real_fct a f)). Unfold continuity; Intros; Apply (continuity_pt_scal f a x (H x)). Qed. Lemma continuity_inv : (f:R->R) (continuity f)->((x:R) ~``(f x)==0``)->(continuity (inv_fct f)). Unfold continuity; Intros; Apply (continuity_pt_inv f x (H x) (H0 x)). Qed. Lemma continuity_div : (f1,f2:R->R) (continuity f1)->(continuity f2)->((x:R) ~``(f2 x)==0``)->(continuity (div_fct f1 f2)). Unfold continuity; Intros; Apply (continuity_pt_div f1 f2 x (H x) (H0 x) (H1 x)). Qed. Lemma continuity_comp : (f1,f2:R->R) (continuity f1) -> (continuity f2) -> (continuity (comp f2 f1)). Unfold continuity; Intros. Apply (continuity_pt_comp f1 f2 x (H x) (H0 (f1 x))). Qed. (*****************************************************) (** Derivative's definition using Landau's kernel *) (*****************************************************) Definition derivable_pt_lim [f:R->R;x,l:R] : Prop := ((eps:R) ``0(EXT delta : posreal | ((h:R) ~``h==0``->``(Rabsolu h) ``(Rabsolu ((((f (x+h))-(f x))/h)-l))R;x:R] : R -> Prop := [l:R](derivable_pt_lim f x l). Definition derivable_pt [f:R->R;x:R] := (SigT R (derivable_pt_abs f x)). Definition derivable [f:R->R] := (x:R)(derivable_pt f x). Definition derive_pt [f:R->R;x:R;pr:(derivable_pt f x)] := (projT1 ? ? pr). Definition derive [f:R->R;pr:(derivable f)] := [x:R](derive_pt f x (pr x)). Arguments Scope derivable_pt_lim [Rfun_scope R_scope]. Arguments Scope derivable_pt_abs [Rfun_scope R_scope R_scope]. Arguments Scope derivable_pt [Rfun_scope R_scope]. Arguments Scope derivable [Rfun_scope]. Arguments Scope derive_pt [Rfun_scope R_scope _]. Arguments Scope derive [Rfun_scope _]. Definition antiderivative [f,g:R->R;a,b:R] : Prop := ((x:R)``a<=x<=b``->(EXT pr : (derivable_pt g x) | (f x)==(derive_pt g x pr)))/\``a<=b``. (************************************) (** 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 cond_D1)) }. (**********) Lemma unicite_step1 : (f:R->R;x,l1,l2:R) (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l1 R0) -> (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l2 R0) -> l1 == l2. Intros; Apply (single_limit [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l1 l2 R0); Try Assumption. Unfold adhDa; Intros; Exists ``alp/2``. Split. Unfold Rdiv; Apply prod_neq_R0. Red; Intro; Rewrite H2 in H1; Elim (Rlt_antirefl ? H1). Apply Rinv_neq_R0; DiscrR. Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Unfold Rdiv; Rewrite Rabsolu_mult. Replace ``(Rabsolu (/2))`` with ``/2``. Replace (Rabsolu alp) with alp. Apply Rlt_monotony_contra with ``2``. Sup0. Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]; Rewrite Rmult_1r; Rewrite double; Pattern 1 alp; Replace alp with ``alp+0``; [Idtac | Ring]; Apply Rlt_compatibility; Assumption. Symmetry; Apply Rabsolu_right; Left; Assumption. Symmetry; Apply Rabsolu_right; Left; Change ``0R;x,l:R) (derivable_pt_lim f x l) -> (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l R0). Unfold derivable_pt_lim; Intros; Unfold limit1_in; Unfold limit_in; Intros. Assert H1 := (H eps H0). Elim H1 ; Intros. Exists (pos x0). Split. Apply (cond_pos x0). Simpl; Unfold R_dist; Intros. Elim H3; Intros. Apply H2; [Assumption |Unfold Rminus in H5; Rewrite Ropp_O in H5; Rewrite Rplus_Or in H5; Assumption]. Qed. Lemma unicite_step3 : (f:R->R;x,l:R) (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l R0) -> (derivable_pt_lim f x l). Unfold limit1_in derivable_pt_lim; Unfold limit_in; Unfold dist; Simpl; Intros. Elim (H eps H0). Intros; Elim H1; Intros. Exists (mkposreal x0 H2). Simpl; Intros; Unfold R_dist in H3; Apply (H3 h). Split; [Assumption | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Assumption]. Qed. Lemma unicite_limite : (f:R->R;x,l1,l2:R) (derivable_pt_lim f x l1) -> (derivable_pt_lim f x l2) -> l1==l2. Intros. Assert H1 := (unicite_step2 ? ? ? H). Assert H2 := (unicite_step2 ? ? ? H0). Assert H3 := (unicite_step1 ? ? ? ? H1 H2). Assumption. Qed. Lemma derive_pt_eq : (f:R->R;x,l:R;pr:(derivable_pt f x)) (derive_pt f x pr)==l <-> (derivable_pt_lim f x l). Intros; Split. Intro; Assert H1 := (projT2 ? ? pr); Unfold derive_pt in H; Rewrite H in H1; Assumption. Intro; Assert H1 := (projT2 ? ? pr); Unfold derivable_pt_abs in H1. Assert H2 := (unicite_limite ? ? ? ? H H1). Unfold derive_pt; Unfold derivable_pt_abs. Symmetry; Assumption. Qed. (**********) Lemma derive_pt_eq_0 : (f:R->R;x,l:R;pr:(derivable_pt f x)) (derivable_pt_lim f x l) -> (derive_pt f x pr)==l. Intros; Elim (derive_pt_eq f x l pr); Intros. Apply (H1 H). Qed. (**********) Lemma derive_pt_eq_1 : (f:R->R;x,l:R;pr:(derivable_pt f x)) (derive_pt f x pr)==l -> (derivable_pt_lim f x l). Intros; Elim (derive_pt_eq f x l pr); Intros. Apply (H0 H). Qed. (********************************************************************) (** Equivalence of this definition with the one using limit concept *) (********************************************************************) Lemma derive_pt_D_in : (f,df:R->R;x:R;pr:(derivable_pt f x)) (D_in f df no_cond x) <-> (derive_pt f x pr)==(df x). Intros; Split. Unfold D_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. Apply derive_pt_eq_0. Unfold derivable_pt_lim. 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. Assert H0 := (derive_pt_eq_1 f x (df x) pr H). 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. Qed. Lemma derivable_pt_lim_D_in : (f,df:R->R;x:R) (D_in f df no_cond x) <-> (derivable_pt_lim f x (df x)). Intros; Split. Unfold D_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. Unfold derivable_pt_lim. 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. Unfold derivable_pt_lim in H. Unfold D_in; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. Elim (H eps H0); Intros alpha H2; Exists (pos alpha); Split. Apply (cond_pos alpha). Intros. Elim H1; Intros; Unfold D_x in H3; Elim H3; Intros; Cut ``x0-x<>0``. Intro; Generalize (H2 ``x0-x`` H7 H4); Replace ``x+(x0-x)`` with x0. Intro; Assumption. Ring. Auto with real. Qed. (***********************************) (** derivability -> continuity *) (***********************************) (**********) Lemma derivable_derive : (f:R->R;x:R;pr:(derivable_pt f x)) (EXT l : R | (derive_pt f x pr)==l). Intros; Exists (projT1 ? ? pr). Unfold derive_pt; Reflexivity. Qed. Theorem derivable_continuous_pt : (f:R->R;x:R) (derivable_pt f x) -> (continuity_pt f x). Intros. Generalize (derivable_derive f x X); Intro. Elim H; Intros l H1. Cut l==((fct_cte l) x). Intro. Rewrite H0 in H1. Generalize (derive_pt_D_in f (fct_cte l) x); Intro. Elim (H2 X); Intros. Generalize (H4 H1); Intro. Unfold continuity_pt. Apply (cont_deriv f (fct_cte l) no_cond x H5). Unfold fct_cte; Reflexivity. Qed. Theorem derivable_continuous : (f:R->R) (derivable f) -> (continuity f). Unfold derivable continuity; Intros. Apply (derivable_continuous_pt f x (X x)). Qed. (****************************************************************) (** Main rules *) (****************************************************************) Lemma derivable_pt_lim_plus : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 x l2) -> (derivable_pt_lim (plus_fct f1 f2) x ``l1+l2``). Intros. Apply unicite_step3. Assert H1 := (unicite_step2 ? ? ? H). Assert H2 := (unicite_step2 ? ? ? H0). Unfold plus_fct. Cut (h:R)``((f1 (x+h))+(f2 (x+h))-((f1 x)+(f2 x)))/h``==``((f1 (x+h))-(f1 x))/h+((f2 (x+h))-(f2 x))/h``. Intro. Generalize(limit_plus [h':R]``((f1 (x+h'))-(f1 x))/h'`` [h':R]``((f2 (x+h'))-(f2 x))/h'`` [h:R]``h <> 0`` l1 l2 ``0`` H1 H2). Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. Elim (H4 eps H5); Intros. Exists x0. Elim H6; Intros. Split. Assumption. Intros; Rewrite H3; Apply H8; Assumption. Intro; Unfold Rdiv; Ring. Qed. Lemma derivable_pt_lim_opp : (f:R->R;x,l:R) (derivable_pt_lim f x l) -> (derivable_pt_lim (opp_fct f) x (Ropp l)). Intros. Apply unicite_step3. Assert H1 := (unicite_step2 ? ? ? H). Unfold opp_fct. Cut (h:R) ``( -(f (x+h))- -(f x))/h``==(Ropp ``((f (x+h))-(f x))/h``). Intro. Generalize (limit_Ropp [h:R]``((f (x+h))-(f x))/h``[h:R]``h <> 0`` l ``0`` H1). Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. Elim (H2 eps H3); Intros. Exists x0. Elim H4; Intros. Split. Assumption. Intros; Rewrite H0; Apply H6; Assumption. Intro; Unfold Rdiv; Ring. Qed. Lemma derivable_pt_lim_minus : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 x l2) -> (derivable_pt_lim (minus_fct f1 f2) x ``l1-l2``). Intros. Apply unicite_step3. Assert H1 := (unicite_step2 ? ? ? H). Assert H2 := (unicite_step2 ? ? ? H0). Unfold minus_fct. Cut (h:R)``((f1 (x+h))-(f1 x))/h-((f2 (x+h))-(f2 x))/h``==``((f1 (x+h))-(f2 (x+h))-((f1 x)-(f2 x)))/h``. Intro. Generalize (limit_minus [h':R]``((f1 (x+h'))-(f1 x))/h'`` [h':R]``((f2 (x+h'))-(f2 x))/h'`` [h:R]``h <> 0`` l1 l2 ``0`` H1 H2). Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. Elim (H4 eps H5); Intros. Exists x0. Elim H6; Intros. Split. Assumption. Intros; Rewrite <- H3; Apply H8; Assumption. Intro; Unfold Rdiv; Ring. Qed. Lemma derivable_pt_lim_mult : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 x l2) -> (derivable_pt_lim (mult_fct f1 f2) x ``l1*(f2 x)+(f1 x)*l2``). Intros. Assert H1 := (derivable_pt_lim_D_in f1 [y:R]l1 x). Elim H1; Intros. Assert H4 := (H3 H). Assert H5 := (derivable_pt_lim_D_in f2 [y:R]l2 x). Elim H5; Intros. Assert H8 := (H7 H0). Clear H1 H2 H3 H5 H6 H7. Assert H1 := (derivable_pt_lim_D_in (mult_fct f1 f2) [y:R]``l1*(f2 x)+(f1 x)*l2`` x). Elim H1; Intros. Clear H1 H3. Apply H2. Unfold mult_fct. Apply (Dmult no_cond [y:R]l1 [y:R]l2 f1 f2 x); Assumption. Qed. Lemma derivable_pt_lim_const : (a,x:R) (derivable_pt_lim (fct_cte a) x ``0``). Intros; Unfold fct_cte derivable_pt_lim. 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. Qed. Lemma derivable_pt_lim_scal : (f:R->R;a,x,l:R) (derivable_pt_lim f x l) -> (derivable_pt_lim (mult_real_fct a f) x ``a*l``). Intros. Assert H0 := (derivable_pt_lim_const a x). Replace (mult_real_fct a f) with (mult_fct (fct_cte a) f). Replace ``a*l`` with ``0*(f x)+a*l``; [Idtac | Ring]. Apply (derivable_pt_lim_mult (fct_cte a) f x ``0`` l); Assumption. Unfold mult_real_fct mult_fct fct_cte; Reflexivity. Qed. Lemma derivable_pt_lim_id : (x:R) (derivable_pt_lim id x ``1``). Intro; Unfold derivable_pt_lim. Intros eps Heps; Exists (mkposreal eps Heps); Intros h H1 H2; Unfold id; Replace ``(x+h-x)/h-1`` with ``0``. Rewrite Rabsolu_R0; Apply Rle_lt_trans with ``(Rabsolu h)``. Apply Rabsolu_pos. Assumption. Unfold Rminus; Rewrite Rplus_assoc; Rewrite (Rplus_sym x); Rewrite Rplus_assoc. Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Unfold Rdiv; Rewrite <- Rinv_r_sym. Symmetry; Apply Rplus_Ropp_r. Assumption. Qed. Lemma derivable_pt_lim_Rsqr : (x:R) (derivable_pt_lim Rsqr x ``2*x``). Intro; Unfold derivable_pt_lim. Unfold Rsqr; Intros eps Heps; Exists (mkposreal eps Heps); Intros h H1 H2; Replace ``((x+h)*(x+h)-x*x)/h-2*x`` with ``h``. Assumption. Replace ``(x+h)*(x+h)-x*x`` with ``2*x*h+h*h``; [Idtac | Ring]. Unfold Rdiv; Rewrite Rmult_Rplus_distrl. Repeat Rewrite Rmult_assoc. Repeat Rewrite <- Rinv_r_sym; [Idtac | Assumption]. Ring. Qed. Lemma derivable_pt_lim_comp : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 (f1 x) l2) -> (derivable_pt_lim (comp f2 f1) x ``l2*l1``). Intros; Assert H1 := (derivable_pt_lim_D_in f1 [y:R]l1 x). Elim H1; Intros. Assert H4 := (H3 H). Assert H5 := (derivable_pt_lim_D_in f2 [y:R]l2 (f1 x)). Elim H5; Intros. Assert H8 := (H7 H0). Clear H1 H2 H3 H5 H6 H7. Assert H1 := (derivable_pt_lim_D_in (comp f2 f1) [y:R]``l2*l1`` x). Elim H1; Intros. Clear H1 H3; Apply H2. Unfold comp; Cut (D_in [x0:R](f2 (f1 x0)) [y:R]``l2*l1`` (Dgf no_cond no_cond f1) x) -> (D_in [x0:R](f2 (f1 x0)) [y:R]``l2*l1`` no_cond x). Intro; Apply H1. Rewrite Rmult_sym; Apply (Dcomp no_cond no_cond [y:R]l1 [y:R]l2 f1 f2 x); Assumption. Unfold Dgf D_in no_cond; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. Elim (H1 eps H3); Intros. Exists x0; Intros; Split. Elim H5; Intros; Assumption. Intros; Elim H5; Intros; Apply H9; Split. Unfold D_x; Split. Split; Trivial. Elim H6; Intros; Unfold D_x in H10; Elim H10; Intros; Assumption. Elim H6; Intros; Assumption. Qed. Lemma derivable_pt_plus : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> (derivable_pt (plus_fct f1 f2) x). Unfold derivable_pt; Intros. Elim X; Intros. Elim X0; Intros. Apply Specif.existT with ``x0+x1``. Apply derivable_pt_lim_plus; Assumption. Qed. Lemma derivable_pt_opp : (f:R->R;x:R) (derivable_pt f x) -> (derivable_pt (opp_fct f) x). Unfold derivable_pt; Intros. Elim X; Intros. Apply Specif.existT with ``-x0``. Apply derivable_pt_lim_opp; Assumption. Qed. Lemma derivable_pt_minus : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> (derivable_pt (minus_fct f1 f2) x). Unfold derivable_pt; Intros. Elim X; Intros. Elim X0; Intros. Apply Specif.existT with ``x0-x1``. Apply derivable_pt_lim_minus; Assumption. Qed. Lemma derivable_pt_mult : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> (derivable_pt (mult_fct f1 f2) x). Unfold derivable_pt; Intros. Elim X; Intros. Elim X0; Intros. Apply Specif.existT with ``x0*(f2 x)+(f1 x)*x1``. Apply derivable_pt_lim_mult; Assumption. Qed. Lemma derivable_pt_const : (a,x:R) (derivable_pt (fct_cte a) x). Intros; Unfold derivable_pt. Apply Specif.existT with ``0``. Apply derivable_pt_lim_const. Qed. Lemma derivable_pt_scal : (f:R->R;a,x:R) (derivable_pt f x) -> (derivable_pt (mult_real_fct a f) x). Unfold derivable_pt; Intros. Elim X; Intros. Apply Specif.existT with ``a*x0``. Apply derivable_pt_lim_scal; Assumption. Qed. Lemma derivable_pt_id : (x:R) (derivable_pt id x). Unfold derivable_pt; Intro. Exists ``1``. Apply derivable_pt_lim_id. Qed. Lemma derivable_pt_Rsqr : (x:R) (derivable_pt Rsqr x). Unfold derivable_pt; Intro; Apply Specif.existT with ``2*x``. Apply derivable_pt_lim_Rsqr. Qed. Lemma derivable_pt_comp : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 (f1 x)) -> (derivable_pt (comp f2 f1) x). Unfold derivable_pt; Intros. Elim X; Intros. Elim X0 ;Intros. Apply Specif.existT with ``x1*x0``. Apply derivable_pt_lim_comp; Assumption. Qed. Lemma derivable_plus : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (plus_fct f1 f2)). Unfold derivable; Intros. Apply (derivable_pt_plus ? ? x (X ?) (X0 ?)). Qed. Lemma derivable_opp : (f:R->R) (derivable f) -> (derivable (opp_fct f)). Unfold derivable; Intros. Apply (derivable_pt_opp ? x (X ?)). Qed. Lemma derivable_minus : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (minus_fct f1 f2)). Unfold derivable; Intros. Apply (derivable_pt_minus ? ? x (X ?) (X0 ?)). Qed. Lemma derivable_mult : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (mult_fct f1 f2)). Unfold derivable; Intros. Apply (derivable_pt_mult ? ? x (X ?) (X0 ?)). Qed. Lemma derivable_const : (a:R) (derivable (fct_cte a)). Unfold derivable; Intros. Apply derivable_pt_const. Qed. Lemma derivable_scal : (f:R->R;a:R) (derivable f) -> (derivable (mult_real_fct a f)). Unfold derivable; Intros. Apply (derivable_pt_scal ? a x (X ?)). Qed. Lemma derivable_id : (derivable id). Unfold derivable; Intro; Apply derivable_pt_id. Qed. Lemma derivable_Rsqr : (derivable Rsqr). Unfold derivable; Intro; Apply derivable_pt_Rsqr. Qed. Lemma derivable_comp : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (comp f2 f1)). Unfold derivable; Intros. Apply (derivable_pt_comp ? ? x (X ?) (X0 ?)). Qed. Lemma derive_pt_plus : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 x)) ``(derive_pt (plus_fct f1 f2) x (derivable_pt_plus ? ? ? pr1 pr2)) == (derive_pt f1 x pr1) + (derive_pt f2 x pr2)``. Intros. Assert H := (derivable_derive f1 x pr1). Assert H0 := (derivable_derive f2 x pr2). Assert H1 := (derivable_derive (plus_fct f1 f2) x (derivable_pt_plus ? ? ? pr1 pr2)). Elim H; Clear H; Intros l1 H. Elim H0; Clear H0; Intros l2 H0. Elim H1; Clear H1; Intros l H1. Rewrite H; Rewrite H0; Apply derive_pt_eq_0. Assert H3 := (projT2 ? ? pr1). Unfold derive_pt in H; Rewrite H in H3. Assert H4 := (projT2 ? ? pr2). Unfold derive_pt in H0; Rewrite H0 in H4. Apply derivable_pt_lim_plus; Assumption. Qed. Lemma derive_pt_opp : (f:R->R;x:R;pr1:(derivable_pt f x)) ``(derive_pt (opp_fct f) x (derivable_pt_opp ? ? pr1)) == -(derive_pt f x pr1)``. Intros. Assert H := (derivable_derive f x pr1). Assert H0 := (derivable_derive (opp_fct f) x (derivable_pt_opp ? ? pr1)). Elim H; Clear H; Intros l1 H. Elim H0; Clear H0; Intros l2 H0. Rewrite H; Apply derive_pt_eq_0. Assert H3 := (projT2 ? ? pr1). Unfold derive_pt in H; Rewrite H in H3. Apply derivable_pt_lim_opp; Assumption. Qed. Lemma derive_pt_minus : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 x)) ``(derive_pt (minus_fct f1 f2) x (derivable_pt_minus ? ? ? pr1 pr2)) == (derive_pt f1 x pr1) - (derive_pt f2 x pr2)``. Intros. Assert H := (derivable_derive f1 x pr1). Assert H0 := (derivable_derive f2 x pr2). Assert H1 := (derivable_derive (minus_fct f1 f2) x (derivable_pt_minus ? ? ? pr1 pr2)). Elim H; Clear H; Intros l1 H. Elim H0; Clear H0; Intros l2 H0. Elim H1; Clear H1; Intros l H1. Rewrite H; Rewrite H0; Apply derive_pt_eq_0. Assert H3 := (projT2 ? ? pr1). Unfold derive_pt in H; Rewrite H in H3. Assert H4 := (projT2 ? ? pr2). Unfold derive_pt in H0; Rewrite H0 in H4. Apply derivable_pt_lim_minus; Assumption. Qed. Lemma derive_pt_mult : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 x)) ``(derive_pt (mult_fct f1 f2) x (derivable_pt_mult ? ? ? pr1 pr2)) == (derive_pt f1 x pr1)*(f2 x) + (f1 x)*(derive_pt f2 x pr2)``. Intros. Assert H := (derivable_derive f1 x pr1). Assert H0 := (derivable_derive f2 x pr2). Assert H1 := (derivable_derive (mult_fct f1 f2) x (derivable_pt_mult ? ? ? pr1 pr2)). Elim H; Clear H; Intros l1 H. Elim H0; Clear H0; Intros l2 H0. Elim H1; Clear H1; Intros l H1. Rewrite H; Rewrite H0; Apply derive_pt_eq_0. Assert H3 := (projT2 ? ? pr1). Unfold derive_pt in H; Rewrite H in H3. Assert H4 := (projT2 ? ? pr2). Unfold derive_pt in H0; Rewrite H0 in H4. Apply derivable_pt_lim_mult; Assumption. Qed. Lemma derive_pt_const : (a,x:R) (derive_pt (fct_cte a) x (derivable_pt_const a x)) == R0. Intros. Apply derive_pt_eq_0. Apply derivable_pt_lim_const. Qed. Lemma derive_pt_scal : (f:R->R;a,x:R;pr:(derivable_pt f x)) ``(derive_pt (mult_real_fct a f) x (derivable_pt_scal ? ? ? pr)) == a * (derive_pt f x pr)``. Intros. Assert H := (derivable_derive f x pr). Assert H0 := (derivable_derive (mult_real_fct a f) x (derivable_pt_scal ? ? ? pr)). Elim H; Clear H; Intros l1 H. Elim H0; Clear H0; Intros l2 H0. Rewrite H; Apply derive_pt_eq_0. Assert H3 := (projT2 ? ? pr). Unfold derive_pt in H; Rewrite H in H3. Apply derivable_pt_lim_scal; Assumption. Qed. Lemma derive_pt_id : (x:R) (derive_pt id x (derivable_pt_id ?))==R1. Intros. Apply derive_pt_eq_0. Apply derivable_pt_lim_id. Qed. Lemma derive_pt_Rsqr : (x:R) (derive_pt Rsqr x (derivable_pt_Rsqr ?)) == ``2*x``. Intros. Apply derive_pt_eq_0. Apply derivable_pt_lim_Rsqr. Qed. Lemma derive_pt_comp : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 (f1 x))) ``(derive_pt (comp f2 f1) x (derivable_pt_comp ? ? ? pr1 pr2)) == (derive_pt f2 (f1 x) pr2) * (derive_pt f1 x pr1)``. Intros. Assert H := (derivable_derive f1 x pr1). Assert H0 := (derivable_derive f2 (f1 x) pr2). Assert H1 := (derivable_derive (comp f2 f1) x (derivable_pt_comp ? ? ? pr1 pr2)). Elim H; Clear H; Intros l1 H. Elim H0; Clear H0; Intros l2 H0. Elim H1; Clear H1; Intros l H1. Rewrite H; Rewrite H0; Apply derive_pt_eq_0. Assert H3 := (projT2 ? ? pr1). Unfold derive_pt in H; Rewrite H in H3. Assert H4 := (projT2 ? ? pr2). Unfold derive_pt in H0; Rewrite H0 in H4. Apply derivable_pt_lim_comp; Assumption. Qed. (* Pow *) Definition pow_fct [n:nat] : R->R := [y:R](pow y n). Lemma derivable_pt_lim_pow_pos : (x:R;n:nat) (lt O n) -> (derivable_pt_lim [y:R](pow y n) x ``(INR n)*(pow x (pred n))``). Intros. Induction n. Elim (lt_n_n ? H). Cut n=O\/(lt O n). Intro; Elim H0; Intro. Rewrite H1; Simpl. Replace [y:R]``y*1`` with (mult_fct id (fct_cte R1)). Replace ``1*1`` with ``1*(fct_cte R1 x)+(id x)*0``. Apply derivable_pt_lim_mult. Apply derivable_pt_lim_id. Apply derivable_pt_lim_const. Unfold fct_cte id; Ring. Reflexivity. Replace [y:R](pow y (S n)) with [y:R]``y*(pow y n)``. Replace (pred (S n)) with n; [Idtac | Reflexivity]. Replace [y:R]``y*(pow y n)`` with (mult_fct id [y:R](pow y n)). Pose f := [y:R](pow y n). Replace ``(INR (S n))*(pow x n)`` with (Rplus (Rmult R1 (f x)) (Rmult (id x) (Rmult (INR n) (pow x (pred n))))). Apply derivable_pt_lim_mult. Apply derivable_pt_lim_id. Unfold f; Apply Hrecn; Assumption. Unfold f. Pattern 1 5 n; Replace n with (S (pred n)). Unfold id; Rewrite S_INR; Simpl. Ring. Symmetry; Apply S_pred with O; Assumption. Unfold mult_fct id; Reflexivity. Reflexivity. Inversion H. Left; Reflexivity. Right. Apply lt_le_trans with (1). Apply lt_O_Sn. Assumption. Qed. Lemma derivable_pt_lim_pow : (x:R; n:nat) (derivable_pt_lim [y:R](pow y n) x ``(INR n)*(pow x (pred n))``). Intros. Induction n. Simpl. Rewrite Rmult_Ol. Replace [_:R]``1`` with (fct_cte R1); [Apply derivable_pt_lim_const | Reflexivity]. Apply derivable_pt_lim_pow_pos. Apply lt_O_Sn. Qed. Lemma derivable_pt_pow : (n:nat;x:R) (derivable_pt [y:R](pow y n) x). Intros; Unfold derivable_pt. Apply Specif.existT with ``(INR n)*(pow x (pred n))``. Apply derivable_pt_lim_pow. Qed. Lemma derivable_pow : (n:nat) (derivable [y:R](pow y n)). Intro; Unfold derivable; Intro; Apply derivable_pt_pow. Qed. Lemma derive_pt_pow : (n:nat;x:R) (derive_pt [y:R](pow y n) x (derivable_pt_pow n x))==``(INR n)*(pow x (pred n))``. Intros; Apply derive_pt_eq_0. Apply derivable_pt_lim_pow. Qed. Lemma pr_nu : (f:R->R;x:R;pr1,pr2:(derivable_pt f x)) (derive_pt f x pr1)==(derive_pt f x pr2). Intros. Unfold derivable_pt in pr1. Unfold derivable_pt in pr2. Elim pr1; Intros. Elim pr2; Intros. Unfold derivable_pt_abs in p. Unfold derivable_pt_abs in p0. Simpl. Apply (unicite_limite f x x0 x1 p p0). Qed. (************************************************************) (** Local extremum's condition *) (************************************************************) Theorem deriv_maximum : (f:R->R;a,b,c:R;pr:(derivable_pt f c)) ``a``c((x:R) ``a``x``(f x)<=(f c)``)->``(derive_pt f c pr)==0``. Intros; Case (total_order R0 (derive_pt f c pr)); Intro. Assert H3 := (derivable_derive f c pr). Elim H3; Intros l H4; Rewrite H4 in H2. Assert H5 := (derive_pt_eq_1 f c l pr H4). Cut ``00``. Intro; Cut ``(Rabsolu (Rmin delta/2 ((b-c)/2)))0``. Intro; Cut ``(Rabsolu (Rmax (-(delta/2)) ((a-c)/2)))R;a,b,c:R;pr:(derivable_pt f c)) ``a``c((x:R) ``a``x``(f c)<=(f x)``)->``(derive_pt f c pr)==0``. Intros. Rewrite <- (Ropp_Ropp (derive_pt f c pr)). Apply eq_RoppO. Rewrite <- (derive_pt_opp f c pr). Cut (x:R)(``a``x``((opp_fct f) x)<=((opp_fct f) c)``). Intro. Apply (deriv_maximum (opp_fct f) a b c (derivable_pt_opp ? ? pr) H H0 H2). Intros; Unfold opp_fct; Apply Rge_Ropp; Apply Rle_sym1. Apply (H1 x H2 H3). Qed. Theorem deriv_constant2 : (f:R->R;a,b,c:R;pr:(derivable_pt f c)) ``a``c((x:R) ``a``x``(f x)==(f c)``)->``(derive_pt f c pr)==0``. Intros. EApply deriv_maximum with a b; Try Assumption. Intros; Right; Apply (H1 x H2 H3). Qed. (**********) Lemma nonneg_derivative_0 : (f:R->R;pr:(derivable f)) (increasing f) -> ((x:R) ``0<=(derive_pt f x (pr x))``). Intros; Unfold increasing in H. Assert H0 := (derivable_derive f x (pr x)). Elim H0; Intros l H1. Rewrite H1; Case (total_order R0 l); Intro. Left; Assumption. Elim H2; Intro. Right; Assumption. Assert H4 := (derive_pt_eq_1 f x l (pr x) H1). Cut ``0< -(l/2)``. Intro; Elim (H4 ``-(l/2)`` H5); Intros delta H6. Cut ``delta/2<>0``/\``0