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Diffstat (limited to 'theories7/Reals/Ranalysis.v')
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diff --git a/theories7/Reals/Ranalysis.v b/theories7/Reals/Ranalysis.v deleted file mode 100644 index d5d84f50..00000000 --- a/theories7/Reals/Ranalysis.v +++ /dev/null @@ -1,477 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(*i $Id: Ranalysis.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) - -Require Rbase. -Require Rfunctions. -Require Rtrigo. -Require SeqSeries. -Require Export Ranalysis1. -Require Export Ranalysis2. -Require Export Ranalysis3. -Require Export Rtopology. -Require Export MVT. -Require Export PSeries_reg. -Require Export Exp_prop. -Require Export Rtrigo_reg. -Require Export Rsqrt_def. -Require Export R_sqrt. -Require Export Rtrigo_calc. -Require Export Rgeom. -Require Export RList. -Require Export Sqrt_reg. -Require Export Ranalysis4. -Require Export Rpower. -V7only [Import R_scope.]. Open Local Scope R_scope. - -Axiom AppVar : R. - -(**********) -Recursive Tactic Definition IntroHypG trm := -Match trm With -|[(plus_fct ?1 ?2)] -> - (Match Context With - |[|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 - |[|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 - | _ -> Idtac) -|[(minus_fct ?1 ?2)] -> - (Match Context With - |[|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 - |[|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 - | _ -> Idtac) -|[(mult_fct ?1 ?2)] -> - (Match Context With - |[|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 - |[|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 - | _ -> Idtac) -|[(div_fct ?1 ?2)] -> Let aux = ?2 In - (Match Context With - |[_:(x0:R)``(aux x0)<>0``|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 - |[_:(x0:R)``(aux x0)<>0``|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 - |[|-(derivable ?)] -> Cut ((x0:R)``(aux x0)<>0``); [Intro; IntroHypG ?1; IntroHypG ?2 | Try Assumption] - |[|-(continuity ?)] -> Cut ((x0:R)``(aux x0)<>0``); [Intro; IntroHypG ?1; IntroHypG ?2 | Try Assumption] - | _ -> Idtac) -|[(comp ?1 ?2)] -> - (Match Context With - |[|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 - |[|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 - | _ -> Idtac) -|[(opp_fct ?1)] -> - (Match Context With - |[|-(derivable ?)] -> IntroHypG ?1 - |[|-(continuity ?)] -> IntroHypG ?1 - | _ -> Idtac) -|[(inv_fct ?1)] -> Let aux = ?1 In - (Match Context With - |[_:(x0:R)``(aux x0)<>0``|-(derivable ?)] -> IntroHypG ?1 - |[_:(x0:R)``(aux x0)<>0``|-(continuity ?)] -> IntroHypG ?1 - |[|-(derivable ?)] -> Cut ((x0:R)``(aux x0)<>0``); [Intro; IntroHypG ?1 | Try Assumption] - |[|-(continuity ?)] -> Cut ((x0:R)``(aux x0)<>0``); [Intro; IntroHypG ?1| Try Assumption] - | _ -> Idtac) -|[cos] -> Idtac -|[sin] -> Idtac -|[cosh] -> Idtac -|[sinh] -> Idtac -|[exp] -> Idtac -|[Rsqr] -> Idtac -|[sqrt] -> Idtac -|[id] -> Idtac -|[(fct_cte ?)] -> Idtac -|[(pow_fct ?)] -> Idtac -|[Rabsolu] -> Idtac -|[?1] -> Let p = ?1 In - (Match Context 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). - -(**********) -Recursive Tactic Definition IntroHypL trm pt := -Match trm With -|[(plus_fct ?1 ?2)] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(continuity_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - | _ -> Idtac) -|[(minus_fct ?1 ?2)] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(continuity_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - | _ -> Idtac) -|[(mult_fct ?1 ?2)] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(continuity_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - | _ -> Idtac) -|[(div_fct ?1 ?2)] -> Let aux = ?2 In - (Match Context With - |[_:``(aux pt)<>0``|-(derivable_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[_:``(aux pt)<>0``|-(continuity_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[_:``(aux pt)<>0``|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[id:(x0:R)``(aux x0)<>0``|-(derivable_pt ? ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt; IntroHypL ?2 pt - |[id:(x0:R)``(aux x0)<>0``|-(continuity_pt ? ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt; IntroHypL ?2 pt - |[id:(x0:R)``(aux x0)<>0``|-(eqT ? (derive_pt ? ? ?) ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(derivable_pt ? ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt; IntroHypL ?2 pt | Try Assumption] - |[|-(continuity_pt ? ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt; IntroHypL ?2 pt | Try Assumption] - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt; IntroHypL ?2 pt | Try Assumption] - | _ -> Idtac) -|[(comp ?1 ?2)] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In IntroHypL ?1 pt_f1; IntroHypL ?2 pt - |[|-(continuity_pt ? ?)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In IntroHypL ?1 pt_f1; IntroHypL ?2 pt - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In IntroHypL ?1 pt_f1; IntroHypL ?2 pt - | _ -> Idtac) -|[(opp_fct ?1)] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> IntroHypL ?1 pt - |[|-(continuity_pt ? ?)] -> IntroHypL ?1 pt - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt - | _ -> Idtac) -|[(inv_fct ?1)] -> Let aux = ?1 In - (Match Context With - |[_:``(aux pt)<>0``|-(derivable_pt ? ?)] -> IntroHypL ?1 pt - |[_:``(aux pt)<>0``|-(continuity_pt ? ?)] -> IntroHypL ?1 pt - |[_:``(aux pt)<>0``|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt - |[id:(x0:R)``(aux x0)<>0``|-(derivable_pt ? ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt - |[id:(x0:R)``(aux x0)<>0``|-(continuity_pt ? ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt - |[id:(x0:R)``(aux x0)<>0``|-(eqT ? (derive_pt ? ? ?) ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt - |[|-(derivable_pt ? ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt | Try Assumption] - |[|-(continuity_pt ? ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt| Try Assumption] - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt | Try Assumption] - | _ -> Idtac) -|[cos] -> Idtac -|[sin] -> Idtac -|[cosh] -> Idtac -|[sinh] -> Idtac -|[exp] -> Idtac -|[Rsqr] -> Idtac -|[id] -> Idtac -|[(fct_cte ?)] -> Idtac -|[(pow_fct ?)] -> Idtac -|[sqrt] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> Cut ``0<pt``; [Intro | Try Assumption] - |[|-(continuity_pt ? ?)] -> Cut ``0<=pt``; [Intro | Try Assumption] - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Cut ``0<pt``; [Intro | Try Assumption] - | _ -> Idtac) -|[Rabsolu] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> Cut ``pt<>0``; [Intro | Try Assumption] - | _ -> Idtac) -|[?1] -> Let p = ?1 In - (Match Context 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] - |[|-(eqT ? (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). - -(**********) -Recursive Tactic Definition IsDiff_pt := -Match Context With - (* fonctions de base *) - [|-(derivable_pt Rsqr ?)] -> Apply derivable_pt_Rsqr -|[|-(derivable_pt id ?1)] -> Apply (derivable_pt_id ?1) -|[|-(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; Apply derivable_pt_pow -|[|-(derivable_pt sqrt ?1)] -> Apply (derivable_pt_sqrt ?1); Assumption Orelse Unfold plus_fct minus_fct opp_fct mult_fct div_fct inv_fct comp id fct_cte pow_fct -|[|-(derivable_pt Rabsolu ?1)] -> Apply (derivable_pt_Rabsolu ?1); Assumption Orelse Unfold plus_fct minus_fct opp_fct mult_fct div_fct inv_fct comp id fct_cte pow_fct - (* regles de differentiabilite *) - (* PLUS *) -|[|-(derivable_pt (plus_fct ?1 ?2) ?3)] -> Apply (derivable_pt_plus ?1 ?2 ?3); IsDiff_pt - (* MOINS *) -|[|-(derivable_pt (minus_fct ?1 ?2) ?3)] -> Apply (derivable_pt_minus ?1 ?2 ?3); IsDiff_pt - (* OPPOSE *) -|[|-(derivable_pt (opp_fct ?1) ?2)] -> Apply (derivable_pt_opp ?1 ?2); IsDiff_pt - (* MULTIPLICATION PAR UN SCALAIRE *) -|[|-(derivable_pt (mult_real_fct ?1 ?2) ?3)] -> Apply (derivable_pt_scal ?2 ?1 ?3); IsDiff_pt - (* MULTIPLICATION *) -|[|-(derivable_pt (mult_fct ?1 ?2) ?3)] -> Apply (derivable_pt_mult ?1 ?2 ?3); IsDiff_pt - (* DIVISION *) - |[|-(derivable_pt (div_fct ?1 ?2) ?3)] -> Apply (derivable_pt_div ?1 ?2 ?3); [IsDiff_pt | IsDiff_pt | Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct comp pow_fct id fct_cte] - (* INVERSION *) - |[|-(derivable_pt (inv_fct ?1) ?2)] -> Apply (derivable_pt_inv ?1 ?2); [Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct comp pow_fct id fct_cte | IsDiff_pt] - (* COMPOSITION *) -|[|-(derivable_pt (comp ?1 ?2) ?3)] -> Apply (derivable_pt_comp ?2 ?1 ?3); IsDiff_pt -|[_:(derivable_pt ?1 ?2)|-(derivable_pt ?1 ?2)] -> Assumption -|[_:(derivable ?1) |- (derivable_pt ?1 ?2)] -> Cut (derivable ?1); [Intro HypDDPT; Apply HypDDPT | Assumption] -|[|-True->(derivable_pt ? ?)] -> Intro HypTruE; Clear HypTruE; IsDiff_pt -| _ -> Try Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct. - -(**********) -Recursive Tactic Definition IsDiff_glob := -Match Context With - (* fonctions de base *) - [|-(derivable Rsqr)] -> 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; Apply derivable_pow - (* regles de differentiabilite *) - (* PLUS *) - |[|-(derivable (plus_fct ?1 ?2))] -> Apply (derivable_plus ?1 ?2); IsDiff_glob - (* MOINS *) - |[|-(derivable (minus_fct ?1 ?2))] -> Apply (derivable_minus ?1 ?2); IsDiff_glob - (* OPPOSE *) - |[|-(derivable (opp_fct ?1))] -> Apply (derivable_opp ?1); IsDiff_glob - (* MULTIPLICATION PAR UN SCALAIRE *) - |[|-(derivable (mult_real_fct ?1 ?2))] -> Apply (derivable_scal ?2 ?1); IsDiff_glob - (* MULTIPLICATION *) - |[|-(derivable (mult_fct ?1 ?2))] -> Apply (derivable_mult ?1 ?2); IsDiff_glob - (* DIVISION *) - |[|-(derivable (div_fct ?1 ?2))] -> Apply (derivable_div ?1 ?2); [IsDiff_glob | IsDiff_glob | Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct] - (* INVERSION *) - |[|-(derivable (inv_fct ?1))] -> Apply (derivable_inv ?1); [Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct | IsDiff_glob] - (* COMPOSITION *) - |[|-(derivable (comp sqrt ?))] -> Unfold derivable; Intro; Try IsDiff_pt - |[|-(derivable (comp Rabsolu ?))] -> Unfold derivable; Intro; Try IsDiff_pt - |[|-(derivable (comp ?1 ?2))] -> Apply (derivable_comp ?2 ?1); IsDiff_glob - |[_:(derivable ?1)|-(derivable ?1)] -> Assumption - |[|-True->(derivable ?)] -> Intro HypTruE; Clear HypTruE; IsDiff_glob - | _ -> Try Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct. - -(**********) -Recursive Tactic Definition IsCont_pt := -Match Context With - (* fonctions de base *) - [|-(continuity_pt Rsqr ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_Rsqr -|[|-(continuity_pt id ?1)] -> Apply derivable_continuous_pt; Apply (derivable_pt_id ?1) -|[|-(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; Apply derivable_continuous_pt; Apply derivable_pt_pow -|[|-(continuity_pt sqrt ?1)] -> Apply continuity_pt_sqrt; Assumption Orelse Unfold plus_fct minus_fct opp_fct mult_fct div_fct inv_fct comp id fct_cte pow_fct -|[|-(continuity_pt Rabsolu ?1)] -> Apply (continuity_Rabsolu ?1) - (* regles de differentiabilite *) - (* PLUS *) -|[|-(continuity_pt (plus_fct ?1 ?2) ?3)] -> Apply (continuity_pt_plus ?1 ?2 ?3); IsCont_pt - (* MOINS *) -|[|-(continuity_pt (minus_fct ?1 ?2) ?3)] -> Apply (continuity_pt_minus ?1 ?2 ?3); IsCont_pt - (* OPPOSE *) -|[|-(continuity_pt (opp_fct ?1) ?2)] -> Apply (continuity_pt_opp ?1 ?2); IsCont_pt - (* MULTIPLICATION PAR UN SCALAIRE *) -|[|-(continuity_pt (mult_real_fct ?1 ?2) ?3)] -> Apply (continuity_pt_scal ?2 ?1 ?3); IsCont_pt - (* MULTIPLICATION *) -|[|-(continuity_pt (mult_fct ?1 ?2) ?3)] -> Apply (continuity_pt_mult ?1 ?2 ?3); IsCont_pt - (* DIVISION *) - |[|-(continuity_pt (div_fct ?1 ?2) ?3)] -> Apply (continuity_pt_div ?1 ?2 ?3); [IsCont_pt | IsCont_pt | Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct comp id fct_cte pow_fct] - (* INVERSION *) - |[|-(continuity_pt (inv_fct ?1) ?2)] -> Apply (continuity_pt_inv ?1 ?2); [IsCont_pt | Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct comp id fct_cte pow_fct] - (* COMPOSITION *) -|[|-(continuity_pt (comp ?1 ?2) ?3)] -> Apply (continuity_pt_comp ?2 ?1 ?3); IsCont_pt -|[_:(continuity_pt ?1 ?2)|-(continuity_pt ?1 ?2)] -> Assumption -|[_:(continuity ?1) |- (continuity_pt ?1 ?2)] -> Cut (continuity ?1); [Intro HypDDPT; Apply HypDDPT | Assumption] -|[_:(derivable_pt ?1 ?2)|-(continuity_pt ?1 ?2)] -> Apply derivable_continuous_pt; Assumption -|[_:(derivable ?1)|-(continuity_pt ?1 ?2)] -> Cut (continuity ?1); [Intro HypDDPT; Apply HypDDPT | Apply derivable_continuous; Assumption] -|[|-True->(continuity_pt ? ?)] -> Intro HypTruE; Clear HypTruE; IsCont_pt -| _ -> Try Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct. - -(**********) -Recursive Tactic Definition IsCont_glob := -Match Context With - (* fonctions de base *) - [|-(continuity Rsqr)] -> 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; Apply derivable_continuous; Apply derivable_pow - |[|-(continuity sinh)] -> Apply derivable_continuous; Apply derivable_sinh - |[|-(continuity cosh)] -> Apply derivable_continuous; Apply derivable_cosh - |[|-(continuity Rabsolu)] -> Apply continuity_Rabsolu - (* regles de continuite *) - (* PLUS *) -|[|-(continuity (plus_fct ?1 ?2))] -> Apply (continuity_plus ?1 ?2); Try IsCont_glob Orelse Assumption - (* MOINS *) -|[|-(continuity (minus_fct ?1 ?2))] -> Apply (continuity_minus ?1 ?2); Try IsCont_glob Orelse Assumption - (* OPPOSE *) -|[|-(continuity (opp_fct ?1))] -> Apply (continuity_opp ?1); Try IsCont_glob Orelse Assumption - (* INVERSE *) -|[|-(continuity (inv_fct ?1))] -> Apply (continuity_inv ?1); Try IsCont_glob Orelse Assumption - (* MULTIPLICATION PAR UN SCALAIRE *) -|[|-(continuity (mult_real_fct ?1 ?2))] -> Apply (continuity_scal ?2 ?1); Try IsCont_glob Orelse Assumption - (* MULTIPLICATION *) -|[|-(continuity (mult_fct ?1 ?2))] -> Apply (continuity_mult ?1 ?2); Try IsCont_glob Orelse Assumption - (* DIVISION *) - |[|-(continuity (div_fct ?1 ?2))] -> Apply (continuity_div ?1 ?2); [Try IsCont_glob Orelse Assumption | Try IsCont_glob Orelse Assumption | Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte pow_fct] - (* COMPOSITION *) - |[|-(continuity (comp sqrt ?))] -> Unfold continuity_pt; Intro; Try IsCont_pt - |[|-(continuity (comp ?1 ?2))] -> Apply (continuity_comp ?2 ?1); Try IsCont_glob Orelse Assumption - |[_:(continuity ?1)|-(continuity ?1)] -> Assumption - |[|-True->(continuity ?)] -> Intro HypTruE; Clear HypTruE; IsCont_glob - |[_:(derivable ?1)|-(continuity ?1)] -> Apply derivable_continuous; Assumption - | _ -> Try Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct. - -(**********) -Recursive Tactic Definition RewTerm trm := -Match trm With -| [(Rplus ?1 ?2)] -> Let p1= (RewTerm ?1) And p2 = (RewTerm ?2) In - (Match p1 With - [(fct_cte ?3)] -> - (Match p2 With - | [(fct_cte ?4)] -> '(fct_cte (Rplus ?3 ?4)) - | _ -> '(plus_fct p1 p2)) - | _ -> '(plus_fct p1 p2)) -| [(Rminus ?1 ?2)] -> Let p1 = (RewTerm ?1) And p2 = (RewTerm ?2) In - (Match p1 With - [(fct_cte ?3)] -> - (Match p2 With - | [(fct_cte ?4)] -> '(fct_cte (Rminus ?3 ?4)) - | _ -> '(minus_fct p1 p2)) - | _ -> '(minus_fct p1 p2)) -| [(Rdiv ?1 ?2)] -> Let p1 = (RewTerm ?1) And p2 = (RewTerm ?2) In - (Match p1 With - [(fct_cte ?3)] -> - (Match p2 With - | [(fct_cte ?4)] -> '(fct_cte (Rdiv ?3 ?4)) - | _ -> '(div_fct p1 p2)) - | _ -> - (Match p2 With - | [(fct_cte ?4)] -> '(mult_fct p1 (fct_cte (Rinv ?4))) - | _ -> '(div_fct p1 p2))) -| [(Rmult ?1 (Rinv ?2))] -> Let p1 = (RewTerm ?1) And p2 = (RewTerm ?2) In - (Match p1 With - [(fct_cte ?3)] -> - (Match p2 With - | [(fct_cte ?4)] -> '(fct_cte (Rdiv ?3 ?4)) - | _ -> '(div_fct p1 p2)) - | _ -> - (Match p2 With - | [(fct_cte ?4)] -> '(mult_fct p1 (fct_cte (Rinv ?4))) - | _ -> '(div_fct p1 p2))) -| [(Rmult ?1 ?2)] -> Let p1 = (RewTerm ?1) And p2 = (RewTerm ?2) In - (Match p1 With - [(fct_cte ?3)] -> - (Match p2 With - | [(fct_cte ?4)] -> '(fct_cte (Rmult ?3 ?4)) - | _ -> '(mult_fct p1 p2)) - | _ -> '(mult_fct p1 p2)) -| [(Ropp ?1)] -> Let p = (RewTerm ?1) In - (Match p With - [(fct_cte ?2)] -> '(fct_cte (Ropp ?2)) - | _ -> '(opp_fct p)) -| [(Rinv ?1)] -> Let p = (RewTerm ?1) In - (Match p With - [(fct_cte ?2)] -> '(fct_cte (Rinv ?2)) - | _ -> '(inv_fct p)) -| [(?1 AppVar)] -> '?1 -| [(?1 ?2)] -> Let p = (RewTerm ?2) In - (Match p With - | [(fct_cte ?3)] -> '(fct_cte (?1 ?3)) - | _ -> '(comp ?1 p)) -| [AppVar] -> 'id -| [(pow AppVar ?1)] -> '(pow_fct ?1) -| [(pow ?1 ?2)] -> Let p = (RewTerm ?1) In - (Match p With - | [(fct_cte ?3)] -> '(fct_cte (pow_fct ?2 ?3)) - | _ -> '(comp (pow_fct ?2) p)) -| [?1]-> '(fct_cte ?1). - -(**********) -Recursive Tactic Definition ConsProof trm pt := -Match trm With -| [(plus_fct ?1 ?2)] -> Let p1 = (ConsProof ?1 pt) And p2 = (ConsProof ?2 pt) In '(derivable_pt_plus ?1 ?2 pt p1 p2) -| [(minus_fct ?1 ?2)] -> Let p1 = (ConsProof ?1 pt) And p2 = (ConsProof ?2 pt) In '(derivable_pt_minus ?1 ?2 pt p1 p2) -| [(mult_fct ?1 ?2)] -> Let p1 = (ConsProof ?1 pt) And p2 = (ConsProof ?2 pt) In '(derivable_pt_mult ?1 ?2 pt p1 p2) -| [(div_fct ?1 ?2)] -> - (Match Context With - |[id:~((?2 pt)==R0) |- ?] -> Let p1 = (ConsProof ?1 pt) And p2 = (ConsProof ?2 pt) In '(derivable_pt_div ?1 ?2 pt p1 p2 id) - | _ -> 'False) -| [(inv_fct ?1)] -> - (Match Context With - |[id:~((?1 pt)==R0) |- ?] -> Let p1 = (ConsProof ?1 pt) In '(derivable_pt_inv ?1 pt p1 id) - | _ -> 'False) -| [(comp ?1 ?2)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In Let p1 = (ConsProof ?1 pt_f1) And p2 = (ConsProof ?2 pt) In '(derivable_pt_comp ?2 ?1 pt p2 p1) -| [(opp_fct ?1)] -> Let p1 = (ConsProof ?1 pt) In '(derivable_pt_opp ?1 pt p1) -| [sin] -> '(derivable_pt_sin pt) -| [cos] -> '(derivable_pt_cos pt) -| [sinh] -> '(derivable_pt_sinh pt) -| [cosh] -> '(derivable_pt_cosh pt) -| [exp] -> '(derivable_pt_exp pt) -| [id] -> '(derivable_pt_id pt) -| [Rsqr] -> '(derivable_pt_Rsqr pt) -| [sqrt] -> - (Match Context With - |[id:(Rlt R0 pt) |- ?] -> '(derivable_pt_sqrt pt id) - | _ -> 'False) -| [(fct_cte ?1)] -> '(derivable_pt_const ?1 pt) -| [?1] -> Let aux = ?1 In - (Match Context With - [ id : (derivable_pt aux pt) |- ?] -> 'id - |[ id : (derivable aux) |- ?] -> '(id pt) - | _ -> 'False). - -(**********) -Recursive Tactic Definition SimplifyDerive trm pt := -Match trm With -| [(plus_fct ?1 ?2)] -> Try Rewrite derive_pt_plus; SimplifyDerive ?1 pt; SimplifyDerive ?2 pt -| [(minus_fct ?1 ?2)] -> Try Rewrite derive_pt_minus; SimplifyDerive ?1 pt; SimplifyDerive ?2 pt -| [(mult_fct ?1 ?2)] -> Try Rewrite derive_pt_mult; SimplifyDerive ?1 pt; SimplifyDerive ?2 pt -| [(div_fct ?1 ?2)] -> Try Rewrite derive_pt_div; SimplifyDerive ?1 pt; SimplifyDerive ?2 pt -| [(comp ?1 ?2)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In Try Rewrite derive_pt_comp; SimplifyDerive ?1 pt_f1; SimplifyDerive ?2 pt -| [(opp_fct ?1)] -> Try Rewrite derive_pt_opp; SimplifyDerive ?1 pt -| [(inv_fct ?1)] -> Try Rewrite derive_pt_inv; SimplifyDerive ?1 pt -| [(fct_cte ?1)] -> 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 -| [?1] -> Let aux = ?1 In - (Match Context With - [ id : (eqT ? (derive_pt aux pt ?2) ?); H : (derivable aux) |- ? ] -> Try Replace (derive_pt aux pt (H pt)) with (derive_pt aux pt ?2); [Rewrite id | Apply pr_nu] - |[ id : (eqT ? (derive_pt aux pt ?2) ?); H : (derivable_pt aux pt) |- ? ] -> Try Replace (derive_pt aux pt H) with (derive_pt aux pt ?2); [Rewrite id | Apply pr_nu] - | _ -> Idtac ) -| _ -> Idtac. - -(**********) -Tactic Definition Reg := -Match Context With -| [|-(derivable_pt ?1 ?2)] -> -Let trm = Eval Cbv Beta in (?1 AppVar) In -Let aux = (RewTerm trm) In IntroHypL aux ?2; Try (Change (derivable_pt aux ?2); IsDiff_pt) Orelse IsDiff_pt -| [|-(derivable ?1)] -> -Let trm = Eval Cbv Beta in (?1 AppVar) In -Let aux = (RewTerm trm) In IntroHypG aux; Try (Change (derivable aux); IsDiff_glob) Orelse IsDiff_glob -| [|-(continuity ?1)] -> -Let trm = Eval Cbv Beta in (?1 AppVar) In -Let aux = (RewTerm trm) In IntroHypG aux; Try (Change (continuity aux); IsCont_glob) Orelse IsCont_glob -| [|-(continuity_pt ?1 ?2)] -> -Let trm = Eval Cbv Beta in (?1 AppVar) In -Let aux = (RewTerm trm) In IntroHypL aux ?2; Try (Change (continuity_pt aux ?2); IsCont_pt) Orelse IsCont_pt -| [|-(eqT ? (derive_pt ?1 ?2 ?3) ?4)] -> -Let trm = Eval Cbv Beta in (?1 AppVar) In -Let aux = (RewTerm trm) In -IntroHypL aux ?2; Let aux2 = (ConsProof aux ?2) In Try (Replace (derive_pt ?1 ?2 ?3) with (derive_pt aux ?2 aux2); [SimplifyDerive aux ?2; Try Unfold plus_fct minus_fct mult_fct div_fct id fct_cte inv_fct opp_fct; Try Ring | Try Apply pr_nu]) Orelse IsDiff_pt. |