--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "Coq.ma".
+
+(*#**********************************************************************)
+
+(* v * The Coq Proof Assistant / The Coq Development Team *)
+
+(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
+
+(* \VV/ *************************************************************)
+
+(* // * This file is distributed under the terms of the *)
+
+(* * GNU Lesser General Public License Version 2.1 *)
+
+(*#**********************************************************************)
+
+(*i $Id: Ranalysis.v,v 1.19 2003/11/29 17:28:37 herbelin Exp $ i*)
+
+include "Reals/Rbase.ma".
+
+include "Reals/Rfunctions.ma".
+
+include "Reals/Rtrigo.ma".
+
+include "Reals/SeqSeries.ma".
+
+include "Reals/Ranalysis1.ma".
+
+include "Reals/Ranalysis2.ma".
+
+include "Reals/Ranalysis3.ma".
+
+include "Reals/Rtopology.ma".
+
+include "Reals/MVT.ma".
+
+include "Reals/PSeries_reg.ma".
+
+include "Reals/Exp_prop.ma".
+
+include "Reals/Rtrigo_reg.ma".
+
+include "Reals/Rsqrt_def.ma".
+
+include "Reals/R_sqrt.ma".
+
+include "Reals/Rtrigo_calc.ma".
+
+include "Reals/Rgeom.ma".
+
+include "Reals/RList.ma".
+
+include "Reals/Sqrt_reg.ma".
+
+include "Reals/Ranalysis4.ma".
+
+include "Reals/Rpower.ma".
+
+(* UNEXPORTED
+Open Local Scope R_scope.
+*)
+
+inline procedural "cic:/Coq/Reals/Ranalysis/AppVar.con".
+
+(*#*********)
+
+(* UNEXPORTED
+Ltac intro_hyp_glob trm :=
+ match constr:trm with
+ | (?X1 + ?X2)%F =>
+ match goal with
+ | |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2
+ | |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2
+ | _ => idtac
+ end
+ | (?X1 - ?X2)%F =>
+ match goal with
+ | |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2
+ | |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2
+ | _ => idtac
+ end
+ | (?X1 * ?X2)%F =>
+ match goal with
+ | |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2
+ | |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2
+ | _ => idtac
+ end
+ | (?X1 / ?X2)%F =>
+ let aux := constr:X2 in
+ match goal with
+ | _:(forall x0:R, aux x0 <> 0) |- (derivable _) =>
+ intro_hyp_glob X1; intro_hyp_glob X2
+ | _:(forall x0:R, aux x0 <> 0) |- (continuity _) =>
+ intro_hyp_glob X1; intro_hyp_glob X2
+ | |- (derivable _) =>
+ cut (forall x0:R, aux x0 <> 0);
+ [ intro; intro_hyp_glob X1; intro_hyp_glob X2 | try assumption ]
+ | |- (continuity _) =>
+ cut (forall x0:R, aux x0 <> 0);
+ [ intro; intro_hyp_glob X1; intro_hyp_glob X2 | try assumption ]
+ | _ => idtac
+ end
+ | (comp ?X1 ?X2) =>
+ match goal with
+ | |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2
+ | |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2
+ | _ => idtac
+ end
+ | (- ?X1)%F =>
+ match goal with
+ | |- (derivable _) => intro_hyp_glob X1
+ | |- (continuity _) => intro_hyp_glob X1
+ | _ => idtac
+ end
+ | (/ ?X1)%F =>
+ let aux := constr:X1 in
+ match goal with
+ | _:(forall x0:R, aux x0 <> 0) |- (derivable _) =>
+ intro_hyp_glob X1
+ | _:(forall x0:R, aux x0 <> 0) |- (continuity _) =>
+ intro_hyp_glob X1
+ | |- (derivable _) =>
+ cut (forall x0:R, aux x0 <> 0);
+ [ intro; intro_hyp_glob X1 | try assumption ]
+ | |- (continuity _) =>
+ cut (forall x0:R, aux x0 <> 0);
+ [ intro; intro_hyp_glob X1 | try assumption ]
+ | _ => idtac
+ end
+ | cos => idtac
+ | sin => idtac
+ | cosh => idtac
+ | sinh => idtac
+ | exp => idtac
+ | Rsqr => idtac
+ | sqrt => idtac
+ | id => idtac
+ | (fct_cte _) => idtac
+ | (pow_fct _) => idtac
+ | Rabs => idtac
+ | ?X1 =>
+ let p := constr:X1 in
+ match goal with
+ | _:(derivable p) |- _ => idtac
+ | |- (derivable p) => idtac
+ | |- (derivable _) =>
+ cut (True -> derivable p);
+ [ intro HYPPD; cut (derivable p);
+ [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
+ | idtac ]
+ | _:(continuity p) |- _ => idtac
+ | |- (continuity p) => idtac
+ | |- (continuity _) =>
+ cut (True -> continuity p);
+ [ intro HYPPD; cut (continuity p);
+ [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
+ | idtac ]
+ | _ => idtac
+ end
+ end.
+*)
+
+(*#*********)
+
+(* UNEXPORTED
+Ltac intro_hyp_pt trm pt :=
+ match constr:trm with
+ | (?X1 + ?X2)%F =>
+ match goal with
+ | |- (derivable_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | |- (continuity_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | |- (derive_pt _ _ _ = _) =>
+ intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | _ => idtac
+ end
+ | (?X1 - ?X2)%F =>
+ match goal with
+ | |- (derivable_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | |- (continuity_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | |- (derive_pt _ _ _ = _) =>
+ intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | _ => idtac
+ end
+ | (?X1 * ?X2)%F =>
+ match goal with
+ | |- (derivable_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | |- (continuity_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | |- (derive_pt _ _ _ = _) =>
+ intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | _ => idtac
+ end
+ | (?X1 / ?X2)%F =>
+ let aux := constr:X2 in
+ match goal with
+ | _:(aux pt <> 0) |- (derivable_pt _ _) =>
+ intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | _:(aux pt <> 0) |- (continuity_pt _ _) =>
+ intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | _:(aux pt <> 0) |- (derive_pt _ _ _ = _) =>
+ intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | id:(forall x0:R, aux x0 <> 0) |- (derivable_pt _ _) =>
+ generalize (id pt); intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | id:(forall x0:R, aux x0 <> 0) |- (continuity_pt _ _) =>
+ generalize (id pt); intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | id:(forall x0:R, aux x0 <> 0) |- (derive_pt _ _ _ = _) =>
+ generalize (id pt); intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+ | |- (derivable_pt _ _) =>
+ cut (aux pt <> 0);
+ [ intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt | try assumption ]
+ | |- (continuity_pt _ _) =>
+ cut (aux pt <> 0);
+ [ intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt | try assumption ]
+ | |- (derive_pt _ _ _ = _) =>
+ cut (aux pt <> 0);
+ [ intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt | try assumption ]
+ | _ => idtac
+ end
+ | (comp ?X1 ?X2) =>
+ match goal with
+ | |- (derivable_pt _ _) =>
+ let pt_f1 := eval cbv beta in (X2 pt) in
+ (intro_hyp_pt X1 pt_f1; intro_hyp_pt X2 pt)
+ | |- (continuity_pt _ _) =>
+ let pt_f1 := eval cbv beta in (X2 pt) in
+ (intro_hyp_pt X1 pt_f1; intro_hyp_pt X2 pt)
+ | |- (derive_pt _ _ _ = _) =>
+ let pt_f1 := eval cbv beta in (X2 pt) in
+ (intro_hyp_pt X1 pt_f1; intro_hyp_pt X2 pt)
+ | _ => idtac
+ end
+ | (- ?X1)%F =>
+ match goal with
+ | |- (derivable_pt _ _) => intro_hyp_pt X1 pt
+ | |- (continuity_pt _ _) => intro_hyp_pt X1 pt
+ | |- (derive_pt _ _ _ = _) => intro_hyp_pt X1 pt
+ | _ => idtac
+ end
+ | (/ ?X1)%F =>
+ let aux := constr:X1 in
+ match goal with
+ | _:(aux pt <> 0) |- (derivable_pt _ _) =>
+ intro_hyp_pt X1 pt
+ | _:(aux pt <> 0) |- (continuity_pt _ _) =>
+ intro_hyp_pt X1 pt
+ | _:(aux pt <> 0) |- (derive_pt _ _ _ = _) =>
+ intro_hyp_pt X1 pt
+ | id:(forall x0:R, aux x0 <> 0) |- (derivable_pt _ _) =>
+ generalize (id pt); intro; intro_hyp_pt X1 pt
+ | id:(forall x0:R, aux x0 <> 0) |- (continuity_pt _ _) =>
+ generalize (id pt); intro; intro_hyp_pt X1 pt
+ | id:(forall x0:R, aux x0 <> 0) |- (derive_pt _ _ _ = _) =>
+ generalize (id pt); intro; intro_hyp_pt X1 pt
+ | |- (derivable_pt _ _) =>
+ cut (aux pt <> 0); [ intro; intro_hyp_pt X1 pt | try assumption ]
+ | |- (continuity_pt _ _) =>
+ cut (aux pt <> 0); [ intro; intro_hyp_pt X1 pt | try assumption ]
+ | |- (derive_pt _ _ _ = _) =>
+ cut (aux pt <> 0); [ intro; intro_hyp_pt X1 pt | try assumption ]
+ | _ => idtac
+ end
+ | cos => idtac
+ | sin => idtac
+ | cosh => idtac
+ | sinh => idtac
+ | exp => idtac
+ | Rsqr => idtac
+ | id => idtac
+ | (fct_cte _) => idtac
+ | (pow_fct _) => idtac
+ | sqrt =>
+ match goal with
+ | |- (derivable_pt _ _) => cut (0 < pt); [ intro | try assumption ]
+ | |- (continuity_pt _ _) =>
+ cut (0 <= pt); [ intro | try assumption ]
+ | |- (derive_pt _ _ _ = _) =>
+ cut (0 < pt); [ intro | try assumption ]
+ | _ => idtac
+ end
+ | Rabs =>
+ match goal with
+ | |- (derivable_pt _ _) =>
+ cut (pt <> 0); [ intro | try assumption ]
+ | _ => idtac
+ end
+ | ?X1 =>
+ let p := constr:X1 in
+ match goal with
+ | _:(derivable_pt p pt) |- _ => idtac
+ | |- (derivable_pt p pt) => idtac
+ | |- (derivable_pt _ _) =>
+ cut (True -> derivable_pt p pt);
+ [ intro HYPPD; cut (derivable_pt p pt);
+ [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
+ | idtac ]
+ | _:(continuity_pt p pt) |- _ => idtac
+ | |- (continuity_pt p pt) => idtac
+ | |- (continuity_pt _ _) =>
+ cut (True -> continuity_pt p pt);
+ [ intro HYPPD; cut (continuity_pt p pt);
+ [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
+ | idtac ]
+ | |- (derive_pt _ _ _ = _) =>
+ cut (True -> derivable_pt p pt);
+ [ intro HYPPD; cut (derivable_pt p pt);
+ [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
+ | idtac ]
+ | _ => idtac
+ end
+ end.
+*)
+
+(*#*********)
+
+(* UNEXPORTED
+Ltac is_diff_pt :=
+ match goal with
+ | |- (derivable_pt Rsqr _) =>
+
+ (* fonctions de base *)
+ apply derivable_pt_Rsqr
+ | |- (derivable_pt id ?X1) => apply (derivable_pt_id X1)
+ | |- (derivable_pt (fct_cte _) _) => apply derivable_pt_const
+ | |- (derivable_pt sin _) => apply derivable_pt_sin
+ | |- (derivable_pt cos _) => apply derivable_pt_cos
+ | |- (derivable_pt sinh _) => apply derivable_pt_sinh
+ | |- (derivable_pt cosh _) => apply derivable_pt_cosh
+ | |- (derivable_pt exp _) => apply derivable_pt_exp
+ | |- (derivable_pt (pow_fct _) _) =>
+ unfold pow_fct in |- *; apply derivable_pt_pow
+ | |- (derivable_pt sqrt ?X1) =>
+ apply (derivable_pt_sqrt X1);
+ assumption ||
+ unfold plus_fct, minus_fct, opp_fct, mult_fct, div_fct, inv_fct,
+ comp, id, fct_cte, pow_fct in |- *
+ | |- (derivable_pt Rabs ?X1) =>
+ apply (Rderivable_pt_abs X1);
+ assumption ||
+ unfold plus_fct, minus_fct, opp_fct, mult_fct, div_fct, inv_fct,
+ comp, id, fct_cte, pow_fct in |- *
+ (* regles de differentiabilite *)
+ (* PLUS *)
+ | |- (derivable_pt (?X1 + ?X2) ?X3) =>
+ apply (derivable_pt_plus X1 X2 X3); is_diff_pt
+ (* MOINS *)
+ | |- (derivable_pt (?X1 - ?X2) ?X3) =>
+ apply (derivable_pt_minus X1 X2 X3); is_diff_pt
+ (* OPPOSE *)
+ | |- (derivable_pt (- ?X1) ?X2) =>
+ apply (derivable_pt_opp X1 X2);
+ is_diff_pt
+ (* MULTIPLICATION PAR UN SCALAIRE *)
+ | |- (derivable_pt (mult_real_fct ?X1 ?X2) ?X3) =>
+ apply (derivable_pt_scal X2 X1 X3); is_diff_pt
+ (* MULTIPLICATION *)
+ | |- (derivable_pt (?X1 * ?X2) ?X3) =>
+ apply (derivable_pt_mult X1 X2 X3); is_diff_pt
+ (* DIVISION *)
+ | |- (derivable_pt (?X1 / ?X2) ?X3) =>
+ apply (derivable_pt_div X1 X2 X3);
+ [ is_diff_pt
+ | is_diff_pt
+ | try
+ assumption ||
+ unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+ comp, pow_fct, id, fct_cte in |- * ]
+ | |- (derivable_pt (/ ?X1) ?X2) =>
+
+ (* INVERSION *)
+ apply (derivable_pt_inv X1 X2);
+ [ assumption ||
+ unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+ comp, pow_fct, id, fct_cte in |- *
+ | is_diff_pt ]
+ | |- (derivable_pt (comp ?X1 ?X2) ?X3) =>
+
+ (* COMPOSITION *)
+ apply (derivable_pt_comp X2 X1 X3); is_diff_pt
+ | _:(derivable_pt ?X1 ?X2) |- (derivable_pt ?X1 ?X2) =>
+ assumption
+ | _:(derivable ?X1) |- (derivable_pt ?X1 ?X2) =>
+ cut (derivable X1); [ intro HypDDPT; apply HypDDPT | assumption ]
+ | |- (True -> derivable_pt _ _) =>
+ intro HypTruE; clear HypTruE; is_diff_pt
+ | _ =>
+ try
+ unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id,
+ fct_cte, comp, pow_fct in |- *
+ end.
+*)
+
+(*#*********)
+
+(* UNEXPORTED
+Ltac is_diff_glob :=
+ match goal with
+ | |- (derivable Rsqr) =>
+ (* fonctions de base *)
+ apply derivable_Rsqr
+ | |- (derivable id) => apply derivable_id
+ | |- (derivable (fct_cte _)) => apply derivable_const
+ | |- (derivable sin) => apply derivable_sin
+ | |- (derivable cos) => apply derivable_cos
+ | |- (derivable cosh) => apply derivable_cosh
+ | |- (derivable sinh) => apply derivable_sinh
+ | |- (derivable exp) => apply derivable_exp
+ | |- (derivable (pow_fct _)) =>
+ unfold pow_fct in |- *;
+ apply derivable_pow
+ (* regles de differentiabilite *)
+ (* PLUS *)
+ | |- (derivable (?X1 + ?X2)) =>
+ apply (derivable_plus X1 X2); is_diff_glob
+ (* MOINS *)
+ | |- (derivable (?X1 - ?X2)) =>
+ apply (derivable_minus X1 X2); is_diff_glob
+ (* OPPOSE *)
+ | |- (derivable (- ?X1)) =>
+ apply (derivable_opp X1);
+ is_diff_glob
+ (* MULTIPLICATION PAR UN SCALAIRE *)
+ | |- (derivable (mult_real_fct ?X1 ?X2)) =>
+ apply (derivable_scal X2 X1); is_diff_glob
+ (* MULTIPLICATION *)
+ | |- (derivable (?X1 * ?X2)) =>
+ apply (derivable_mult X1 X2); is_diff_glob
+ (* DIVISION *)
+ | |- (derivable (?X1 / ?X2)) =>
+ apply (derivable_div X1 X2);
+ [ is_diff_glob
+ | is_diff_glob
+ | try
+ assumption ||
+ unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+ id, fct_cte, comp, pow_fct in |- * ]
+ | |- (derivable (/ ?X1)) =>
+
+ (* INVERSION *)
+ apply (derivable_inv X1);
+ [ try
+ assumption ||
+ unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+ id, fct_cte, comp, pow_fct in |- *
+ | is_diff_glob ]
+ | |- (derivable (comp sqrt _)) =>
+
+ (* COMPOSITION *)
+ unfold derivable in |- *; intro; try is_diff_pt
+ | |- (derivable (comp Rabs _)) =>
+ unfold derivable in |- *; intro; try is_diff_pt
+ | |- (derivable (comp ?X1 ?X2)) =>
+ apply (derivable_comp X2 X1); is_diff_glob
+ | _:(derivable ?X1) |- (derivable ?X1) => assumption
+ | |- (True -> derivable _) =>
+ intro HypTruE; clear HypTruE; is_diff_glob
+ | _ =>
+ try
+ unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id,
+ fct_cte, comp, pow_fct in |- *
+ end.
+*)
+
+(*#*********)
+
+(* UNEXPORTED
+Ltac is_cont_pt :=
+ match goal with
+ | |- (continuity_pt Rsqr _) =>
+
+ (* fonctions de base *)
+ apply derivable_continuous_pt; apply derivable_pt_Rsqr
+ | |- (continuity_pt id ?X1) =>
+ apply derivable_continuous_pt; apply (derivable_pt_id X1)
+ | |- (continuity_pt (fct_cte _) _) =>
+ apply derivable_continuous_pt; apply derivable_pt_const
+ | |- (continuity_pt sin _) =>
+ apply derivable_continuous_pt; apply derivable_pt_sin
+ | |- (continuity_pt cos _) =>
+ apply derivable_continuous_pt; apply derivable_pt_cos
+ | |- (continuity_pt sinh _) =>
+ apply derivable_continuous_pt; apply derivable_pt_sinh
+ | |- (continuity_pt cosh _) =>
+ apply derivable_continuous_pt; apply derivable_pt_cosh
+ | |- (continuity_pt exp _) =>
+ apply derivable_continuous_pt; apply derivable_pt_exp
+ | |- (continuity_pt (pow_fct _) _) =>
+ unfold pow_fct in |- *; apply derivable_continuous_pt;
+ apply derivable_pt_pow
+ | |- (continuity_pt sqrt ?X1) =>
+ apply continuity_pt_sqrt;
+ assumption ||
+ unfold plus_fct, minus_fct, opp_fct, mult_fct, div_fct, inv_fct,
+ comp, id, fct_cte, pow_fct in |- *
+ | |- (continuity_pt Rabs ?X1) =>
+ apply (Rcontinuity_abs X1)
+ (* regles de differentiabilite *)
+ (* PLUS *)
+ | |- (continuity_pt (?X1 + ?X2) ?X3) =>
+ apply (continuity_pt_plus X1 X2 X3); is_cont_pt
+ (* MOINS *)
+ | |- (continuity_pt (?X1 - ?X2) ?X3) =>
+ apply (continuity_pt_minus X1 X2 X3); is_cont_pt
+ (* OPPOSE *)
+ | |- (continuity_pt (- ?X1) ?X2) =>
+ apply (continuity_pt_opp X1 X2);
+ is_cont_pt
+ (* MULTIPLICATION PAR UN SCALAIRE *)
+ | |- (continuity_pt (mult_real_fct ?X1 ?X2) ?X3) =>
+ apply (continuity_pt_scal X2 X1 X3); is_cont_pt
+ (* MULTIPLICATION *)
+ | |- (continuity_pt (?X1 * ?X2) ?X3) =>
+ apply (continuity_pt_mult X1 X2 X3); is_cont_pt
+ (* DIVISION *)
+ | |- (continuity_pt (?X1 / ?X2) ?X3) =>
+ apply (continuity_pt_div X1 X2 X3);
+ [ is_cont_pt
+ | is_cont_pt
+ | try
+ assumption ||
+ unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+ comp, id, fct_cte, pow_fct in |- * ]
+ | |- (continuity_pt (/ ?X1) ?X2) =>
+
+ (* INVERSION *)
+ apply (continuity_pt_inv X1 X2);
+ [ is_cont_pt
+ | assumption ||
+ unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+ comp, id, fct_cte, pow_fct in |- * ]
+ | |- (continuity_pt (comp ?X1 ?X2) ?X3) =>
+
+ (* COMPOSITION *)
+ apply (continuity_pt_comp X2 X1 X3); is_cont_pt
+ | _:(continuity_pt ?X1 ?X2) |- (continuity_pt ?X1 ?X2) =>
+ assumption
+ | _:(continuity ?X1) |- (continuity_pt ?X1 ?X2) =>
+ cut (continuity X1); [ intro HypDDPT; apply HypDDPT | assumption ]
+ | _:(derivable_pt ?X1 ?X2) |- (continuity_pt ?X1 ?X2) =>
+ apply derivable_continuous_pt; assumption
+ | _:(derivable ?X1) |- (continuity_pt ?X1 ?X2) =>
+ cut (continuity X1);
+ [ intro HypDDPT; apply HypDDPT
+ | apply derivable_continuous; assumption ]
+ | |- (True -> continuity_pt _ _) =>
+ intro HypTruE; clear HypTruE; is_cont_pt
+ | _ =>
+ try
+ unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id,
+ fct_cte, comp, pow_fct in |- *
+ end.
+*)
+
+(*#*********)
+
+(* UNEXPORTED
+Ltac is_cont_glob :=
+ match goal with
+ | |- (continuity Rsqr) =>
+
+ (* fonctions de base *)
+ apply derivable_continuous; apply derivable_Rsqr
+ | |- (continuity id) => apply derivable_continuous; apply derivable_id
+ | |- (continuity (fct_cte _)) =>
+ apply derivable_continuous; apply derivable_const
+ | |- (continuity sin) => apply derivable_continuous; apply derivable_sin
+ | |- (continuity cos) => apply derivable_continuous; apply derivable_cos
+ | |- (continuity exp) => apply derivable_continuous; apply derivable_exp
+ | |- (continuity (pow_fct _)) =>
+ unfold pow_fct in |- *; apply derivable_continuous; apply derivable_pow
+ | |- (continuity sinh) =>
+ apply derivable_continuous; apply derivable_sinh
+ | |- (continuity cosh) =>
+ apply derivable_continuous; apply derivable_cosh
+ | |- (continuity Rabs) =>
+ apply Rcontinuity_abs
+ (* regles de continuite *)
+ (* PLUS *)
+ | |- (continuity (?X1 + ?X2)) =>
+ apply (continuity_plus X1 X2);
+ try is_cont_glob || assumption
+ (* MOINS *)
+ | |- (continuity (?X1 - ?X2)) =>
+ apply (continuity_minus X1 X2);
+ try is_cont_glob || assumption
+ (* OPPOSE *)
+ | |- (continuity (- ?X1)) =>
+ apply (continuity_opp X1); try is_cont_glob || assumption
+ (* INVERSE *)
+ | |- (continuity (/ ?X1)) =>
+ apply (continuity_inv X1);
+ try is_cont_glob || assumption
+ (* MULTIPLICATION PAR UN SCALAIRE *)
+ | |- (continuity (mult_real_fct ?X1 ?X2)) =>
+ apply (continuity_scal X2 X1);
+ try is_cont_glob || assumption
+ (* MULTIPLICATION *)
+ | |- (continuity (?X1 * ?X2)) =>
+ apply (continuity_mult X1 X2);
+ try is_cont_glob || assumption
+ (* DIVISION *)
+ | |- (continuity (?X1 / ?X2)) =>
+ apply (continuity_div X1 X2);
+ [ try is_cont_glob || assumption
+ | try is_cont_glob || assumption
+ | try
+ assumption ||
+ unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+ id, fct_cte, pow_fct in |- * ]
+ | |- (continuity (comp sqrt _)) =>
+
+ (* COMPOSITION *)
+ unfold continuity_pt in |- *; intro; try is_cont_pt
+ | |- (continuity (comp ?X1 ?X2)) =>
+ apply (continuity_comp X2 X1); try is_cont_glob || assumption
+ | _:(continuity ?X1) |- (continuity ?X1) => assumption
+ | |- (True -> continuity _) =>
+ intro HypTruE; clear HypTruE; is_cont_glob
+ | _:(derivable ?X1) |- (continuity ?X1) =>
+ apply derivable_continuous; assumption
+ | _ =>
+ try
+ unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id,
+ fct_cte, comp, pow_fct in |- *
+ end.
+*)
+
+(*#*********)
+
+(* UNEXPORTED
+Ltac rew_term trm :=
+ match constr:trm with
+ | (?X1 + ?X2) =>
+ let p1 := rew_term X1 with p2 := rew_term X2 in
+ match constr:p1 with
+ | (fct_cte ?X3) =>
+ match constr:p2 with
+ | (fct_cte ?X4) => constr:(fct_cte (X3 + X4))
+ | _ => constr:(p1 + p2)%F
+ end
+ | _ => constr:(p1 + p2)%F
+ end
+ | (?X1 - ?X2) =>
+ let p1 := rew_term X1 with p2 := rew_term X2 in
+ match constr:p1 with
+ | (fct_cte ?X3) =>
+ match constr:p2 with
+ | (fct_cte ?X4) => constr:(fct_cte (X3 - X4))
+ | _ => constr:(p1 - p2)%F
+ end
+ | _ => constr:(p1 - p2)%F
+ end
+ | (?X1 / ?X2) =>
+ let p1 := rew_term X1 with p2 := rew_term X2 in
+ match constr:p1 with
+ | (fct_cte ?X3) =>
+ match constr:p2 with
+ | (fct_cte ?X4) => constr:(fct_cte (X3 / X4))
+ | _ => constr:(p1 / p2)%F
+ end
+ | _ =>
+ match constr:p2 with
+ | (fct_cte ?X4) => constr:(p1 * fct_cte (/ X4))%F
+ | _ => constr:(p1 / p2)%F
+ end
+ end
+ | (?X1 * / ?X2) =>
+ let p1 := rew_term X1 with p2 := rew_term X2 in
+ match constr:p1 with
+ | (fct_cte ?X3) =>
+ match constr:p2 with
+ | (fct_cte ?X4) => constr:(fct_cte (X3 / X4))
+ | _ => constr:(p1 / p2)%F
+ end
+ | _ =>
+ match constr:p2 with
+ | (fct_cte ?X4) => constr:(p1 * fct_cte (/ X4))%F
+ | _ => constr:(p1 / p2)%F
+ end
+ end
+ | (?X1 * ?X2) =>
+ let p1 := rew_term X1 with p2 := rew_term X2 in
+ match constr:p1 with
+ | (fct_cte ?X3) =>
+ match constr:p2 with
+ | (fct_cte ?X4) => constr:(fct_cte (X3 * X4))
+ | _ => constr:(p1 * p2)%F
+ end
+ | _ => constr:(p1 * p2)%F
+ end
+ | (- ?X1) =>
+ let p := rew_term X1 in
+ match constr:p with
+ | (fct_cte ?X2) => constr:(fct_cte (- X2))
+ | _ => constr:(- p)%F
+ end
+ | (/ ?X1) =>
+ let p := rew_term X1 in
+ match constr:p with
+ | (fct_cte ?X2) => constr:(fct_cte (/ X2))
+ | _ => constr:(/ p)%F
+ end
+ | (?X1 AppVar) => constr:X1
+ | (?X1 ?X2) =>
+ let p := rew_term X2 in
+ match constr:p with
+ | (fct_cte ?X3) => constr:(fct_cte (X1 X3))
+ | _ => constr:(comp X1 p)
+ end
+ | AppVar => constr:id
+ | (AppVar ^ ?X1) => constr:(pow_fct X1)
+ | (?X1 ^ ?X2) =>
+ let p := rew_term X1 in
+ match constr:p with
+ | (fct_cte ?X3) => constr:(fct_cte (pow_fct X2 X3))
+ | _ => constr:(comp (pow_fct X2) p)
+ end
+ | ?X1 => constr:(fct_cte X1)
+ end.
+*)
+
+(*#*********)
+
+(* UNEXPORTED
+Ltac deriv_proof trm pt :=
+ match constr:trm with
+ | (?X1 + ?X2)%F =>
+ let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in
+ constr:(derivable_pt_plus X1 X2 pt p1 p2)
+ | (?X1 - ?X2)%F =>
+ let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in
+ constr:(derivable_pt_minus X1 X2 pt p1 p2)
+ | (?X1 * ?X2)%F =>
+ let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in
+ constr:(derivable_pt_mult X1 X2 pt p1 p2)
+ | (?X1 / ?X2)%F =>
+ match goal with
+ | id:(?X2 pt <> 0) |- _ =>
+ let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in
+ constr:(derivable_pt_div X1 X2 pt p1 p2 id)
+ | _ => constr:False
+ end
+ | (/ ?X1)%F =>
+ match goal with
+ | id:(?X1 pt <> 0) |- _ =>
+ let p1 := deriv_proof X1 pt in
+ constr:(derivable_pt_inv X1 pt p1 id)
+ | _ => constr:False
+ end
+ | (comp ?X1 ?X2) =>
+ let pt_f1 := eval cbv beta in (X2 pt) in
+ let p1 := deriv_proof X1 pt_f1 with p2 := deriv_proof X2 pt in
+ constr:(derivable_pt_comp X2 X1 pt p2 p1)
+ | (- ?X1)%F =>
+ let p1 := deriv_proof X1 pt in
+ constr:(derivable_pt_opp X1 pt p1)
+ | sin => constr:(derivable_pt_sin pt)
+ | cos => constr:(derivable_pt_cos pt)
+ | sinh => constr:(derivable_pt_sinh pt)
+ | cosh => constr:(derivable_pt_cosh pt)
+ | exp => constr:(derivable_pt_exp pt)
+ | id => constr:(derivable_pt_id pt)
+ | Rsqr => constr:(derivable_pt_Rsqr pt)
+ | sqrt =>
+ match goal with
+ | id:(0 < pt) |- _ => constr:(derivable_pt_sqrt pt id)
+ | _ => constr:False
+ end
+ | (fct_cte ?X1) => constr:(derivable_pt_const X1 pt)
+ | ?X1 =>
+ let aux := constr:X1 in
+ match goal with
+ | id:(derivable_pt aux pt) |- _ => constr:id
+ | id:(derivable aux) |- _ => constr:(id pt)
+ | _ => constr:False
+ end
+ end.
+*)
+
+(*#*********)
+
+(* UNEXPORTED
+Ltac simplify_derive trm pt :=
+ match constr:trm with
+ | (?X1 + ?X2)%F =>
+ try rewrite derive_pt_plus; simplify_derive X1 pt;
+ simplify_derive X2 pt
+ | (?X1 - ?X2)%F =>
+ try rewrite derive_pt_minus; simplify_derive X1 pt;
+ simplify_derive X2 pt
+ | (?X1 * ?X2)%F =>
+ try rewrite derive_pt_mult; simplify_derive X1 pt;
+ simplify_derive X2 pt
+ | (?X1 / ?X2)%F =>
+ try rewrite derive_pt_div; simplify_derive X1 pt; simplify_derive X2 pt
+ | (comp ?X1 ?X2) =>
+ let pt_f1 := eval cbv beta in (X2 pt) in
+ (try rewrite derive_pt_comp; simplify_derive X1 pt_f1;
+ simplify_derive X2 pt)
+ | (- ?X1)%F => try rewrite derive_pt_opp; simplify_derive X1 pt
+ | (/ ?X1)%F =>
+ try rewrite derive_pt_inv; simplify_derive X1 pt
+ | (fct_cte ?X1) => try rewrite derive_pt_const
+ | id => try rewrite derive_pt_id
+ | sin => try rewrite derive_pt_sin
+ | cos => try rewrite derive_pt_cos
+ | sinh => try rewrite derive_pt_sinh
+ | cosh => try rewrite derive_pt_cosh
+ | exp => try rewrite derive_pt_exp
+ | Rsqr => try rewrite derive_pt_Rsqr
+ | sqrt => try rewrite derive_pt_sqrt
+ | ?X1 =>
+ let aux := constr:X1 in
+ match goal with
+ | id:(derive_pt aux pt ?X2 = _),H:(derivable aux) |- _ =>
+ try replace (derive_pt aux pt (H pt)) with (derive_pt aux pt X2);
+ [ rewrite id | apply pr_nu ]
+ | id:(derive_pt aux pt ?X2 = _),H:(derivable_pt aux pt) |- _ =>
+ try replace (derive_pt aux pt H) with (derive_pt aux pt X2);
+ [ rewrite id | apply pr_nu ]
+ | _ => idtac
+ end
+ | _ => idtac
+ end.
+*)
+
+(*#*********)
+
+(* UNEXPORTED
+Ltac reg :=
+ match goal with
+ | |- (derivable_pt ?X1 ?X2) =>
+ let trm := eval cbv beta in (X1 AppVar) in
+ let aux := rew_term trm in
+ (intro_hyp_pt aux X2;
+ try (change (derivable_pt aux X2) in |- *; is_diff_pt) || is_diff_pt)
+ | |- (derivable ?X1) =>
+ let trm := eval cbv beta in (X1 AppVar) in
+ let aux := rew_term trm in
+ (intro_hyp_glob aux;
+ try (change (derivable aux) in |- *; is_diff_glob) || is_diff_glob)
+ | |- (continuity ?X1) =>
+ let trm := eval cbv beta in (X1 AppVar) in
+ let aux := rew_term trm in
+ (intro_hyp_glob aux;
+ try (change (continuity aux) in |- *; is_cont_glob) || is_cont_glob)
+ | |- (continuity_pt ?X1 ?X2) =>
+ let trm := eval cbv beta in (X1 AppVar) in
+ let aux := rew_term trm in
+ (intro_hyp_pt aux X2;
+ try (change (continuity_pt aux X2) in |- *; is_cont_pt) || is_cont_pt)
+ | |- (derive_pt ?X1 ?X2 ?X3 = ?X4) =>
+ let trm := eval cbv beta in (X1 AppVar) in
+ let aux := rew_term trm in
+ (intro_hyp_pt aux X2;
+ let aux2 := deriv_proof aux X2 in
+ (try
+ (replace (derive_pt X1 X2 X3) with (derive_pt aux X2 aux2);
+ [ simplify_derive aux X2;
+ try
+ unfold plus_fct, minus_fct, mult_fct, div_fct, id, fct_cte,
+ inv_fct, opp_fct in |- *; try ring
+ | try apply pr_nu ]) || is_diff_pt))
+ end.
+*)
+
projects / helm.git / blobdiff