X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fcontribs%2FCoRN-Decl%2Falgebra%2FCPolynomials.ma;fp=matita%2Fcontribs%2FCoRN-Decl%2Falgebra%2FCPolynomials.ma;h=bd621debbd884352c63447ded1d5b6a8ecf78fa8;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hp=0000000000000000000000000000000000000000;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1;p=helm.git

diff --git a/matita/contribs/CoRN-Decl/algebra/CPolynomials.ma b/matita/contribs/CoRN-Decl/algebra/CPolynomials.ma
new file mode 100644
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@@ -0,0 +1,643 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 *********************)
+
+set "baseuri" "cic:/matita/CoRN-Decl/algebra/CPolynomials".
+
+include "CoRN.ma".
+
+(* $Id: CPolynomials.v,v 1.9 2004/04/23 10:00:53 lcf Exp $ *)
+
+(*#* printing _X_ %\ensuremath{x}% *)
+
+(*#* printing _C_ %\ensuremath\diamond% *)
+
+(*#* printing [+X*] %\ensuremath{+x\times}% #+x×# *)
+
+(*#* printing RX %\ensuremath{R[x]}% #R[x]# *)
+
+(*#* printing FX %\ensuremath{F[x]}% #F[x]# *)
+
+include "tactics/RingReflection.ma".
+
+(*#* * Polynomials
+The first section only proves the polynomials form a ring, and nothing more
+interesting.
+Section%~\ref{section:poly-equality}% gives some basic properties of
+equality and induction of polynomials.
+** Definition of polynomials; they form a ring
+%\label{section:poly-ring}%
+*)
+
+(* UNEXPORTED
+Section CPoly_CRing
+*)
+
+(*#*
+%\begin{convention}% Let [CR] be a ring.
+%\end{convention}%
+*)
+
+alias id "CR" = "cic:/CoRN/algebra/CPolynomials/CPoly_CRing/CR.var".
+
+(*#*
+The intuition behind the type [cpoly] is the following
+- [(cpoly CR)] is $CR[X]$ #CR[X]#;
+- [cpoly_zero] is the `empty' polynomial with no coefficients;
+- [(cpoly_linear c p)] is [c[+]X[*]p]
+
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly.ind".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_constant.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_one.con".
+
+(*#*
+Some useful induction lemmas for doubly quantified propositions.
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_ind0.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_sym_ind0.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_ind0'.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_ind0.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_sym_ind0.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_ind0'.con".
+
+(*#* *** The polynomials form a setoid
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_eq_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_eq.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_eq_p_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_ap_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_ap.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_ap_p_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/irreflexive_cpoly_ap.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/symmetric_cpoly_ap.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cotransitive_cpoly_ap.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/tight_apart_cpoly_ap.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_is_CSetoid.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_csetoid.con".
+
+(*#*
+Now that we know that the polynomials form a setoid, we can use the
+notation with [ [#] ] and [ [=] ]. In order to use this notation,
+we introduce [cpoly_zero_cs] and [cpoly_linear_cs], so that Coq
+recognizes we are talking about a setoid.
+We formulate the induction properties and
+the most basic properties of equality and apartness
+in terms of these generators.
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/CPoly_CRing/cpoly_zero_cs.con" "CPoly_CRing__".
+
+inline "cic:/CoRN/algebra/CPolynomials/CPoly_CRing/cpoly_linear_cs.con" "CPoly_CRing__".
+
+inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_ind_cs.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_ind0_cs.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_sym_ind0_cs.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_ind_cs.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_ind0_cs.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_sym_ind0_cs.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_eq_zero_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_cpoly_lin_eq_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_zero_eq_lin_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_cpoly_zero_eq_lin.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_eq_lin_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_cpoly_lin_eq_lin.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_ap_zero_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_cpoly_lin_ap_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_ap_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_zero_ap_lin_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_cpoly_zero_ap_lin.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_zero_ap_lin.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_ap_lin_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_cpoly_lin_ap_lin.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_ap_lin.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_linear_strext.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_linear_wd.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_linear_fun.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_comp_ind.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_triple_comp_ind.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_comp_ind.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_triple_comp_ind.con".
+
+(*#*
+*** The polynomials form a semi-group and a monoid
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_cs.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_zero_plus.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_plus_lin.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_commutative.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_q_ap_q.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_p_plus_ap_p.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_ap_zero_plus.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_op_strext.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_op_wd.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_op.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_plus_associative.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_csemi_grp.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_cm_proof.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_cmonoid.con".
+
+(*#* *** The polynomials form a group
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv_cs.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv_lin.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv_op_strext.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv_op_wd.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_inv_op.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_cg_proof.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_cgroup.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_cag_proof.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_cabgroup.con".
+
+(*#* *** The polynomials form a ring
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_cs.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_zero_mult_cr.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_mult_cr.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_strext.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_wd.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cs.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_zero_mult.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin_mult.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_op_strext.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_op_wd.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_op.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_dist.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_cr_dist.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_assoc_mult_cr.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_assoc_mult.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_lin.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_commutative.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_dist_rht.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_assoc.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_cr_one.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_one_mult.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_one.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_mult_monoid.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_cr_non_triv.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_is_CRing.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_cring.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_constant_strext.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_constant_wd.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_C_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_X_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_x_minus_c.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_x_minus_c_strext.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_x_minus_c_wd.con".
+
+(* UNEXPORTED
+End CPoly_CRing
+*)
+
+(* UNEXPORTED
+Implicit Arguments _C_ [CR].
+*)
+
+(* UNEXPORTED
+Implicit Arguments _X_ [CR].
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_linear_fun'.con".
+
+(* UNEXPORTED
+Implicit Arguments cpoly_linear_fun' [CR].
+*)
+
+(* NOTATION
+Infix "[+X*]" := cpoly_linear_fun' (at level 50, left associativity).
+*)
+
+(*#* ** Apartness, equality, and induction
+%\label{section:poly-equality}%
+*)
+
+(* UNEXPORTED
+Section CPoly_CRing_ctd
+*)
+
+(*#*
+%\begin{convention}%
+Let [CR] be a ring, [p] and [q] polynomials over that ring, and [c] and [d]
+elements of the ring.
+%\end{convention}%
+*)
+
+alias id "CR" = "cic:/CoRN/algebra/CPolynomials/CPoly_CRing_ctd/CR.var".
+
+(* NOTATION
+Notation RX := (cpoly_cring CR).
+*)
+
+(* UNEXPORTED
+Section helpful_section
+*)
+
+alias id "p" = "cic:/CoRN/algebra/CPolynomials/CPoly_CRing_ctd/helpful_section/p.var".
+
+alias id "q" = "cic:/CoRN/algebra/CPolynomials/CPoly_CRing_ctd/helpful_section/q.var".
+
+alias id "c" = "cic:/CoRN/algebra/CPolynomials/CPoly_CRing_ctd/helpful_section/c.var".
+
+alias id "d" = "cic:/CoRN/algebra/CPolynomials/CPoly_CRing_ctd/helpful_section/d.var".
+
+inline "cic:/CoRN/algebra/CPolynomials/linear_eq_zero_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_linear_eq_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/zero_eq_linear_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_zero_eq_linear.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/linear_eq_linear_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_linear_eq_linear.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/linear_ap_zero_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_linear_ap_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/linear_ap_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/zero_ap_linear_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_zero_ap_linear.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/zero_ap_linear.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/linear_ap_linear_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_linear_ap_linear.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/linear_ap_linear.con".
+
+(* UNEXPORTED
+End helpful_section
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_induc.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/Ccpoly_double_sym_ind.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/Cpoly_double_comp_ind.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/Cpoly_triple_comp_ind.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_induc.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_sym_ind.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/poly_double_comp_ind.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/poly_triple_comp_ind.con".
+
+(* UNEXPORTED
+Transparent cpoly_cring.
+*)
+
+(* UNEXPORTED
+Transparent cpoly_cgroup.
+*)
+
+(* UNEXPORTED
+Transparent cpoly_csetoid.
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_apply.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_apply_strext.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_apply_wd.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_apply_fun.con".
+
+(* UNEXPORTED
+End CPoly_CRing_ctd
+*)
+
+(*#*
+%\begin{convention}%
+[cpoly_apply_fun] is denoted infix by [!]
+The first argument is left implicit, so the application of
+polynomial [f] (seen as a function) to argument [x] can be written as [f!x].
+In the names of lemmas, we write [apply].
+%\end{convention}%
+*)
+
+(* UNEXPORTED
+Implicit Arguments cpoly_apply_fun [CR].
+*)
+
+(* NOTATION
+Infix "!" := cpoly_apply_fun (at level 1, no associativity).
+*)
+
+(*#*
+** Basic properties of polynomials
+%\begin{convention}%
+Let [R] be a ring and write [RX] for the ring of polynomials over [R].
+%\end{convention}%
+*)
+
+(* UNEXPORTED
+Section Poly_properties
+*)
+
+alias id "R" = "cic:/CoRN/algebra/CPolynomials/Poly_properties/R.var".
+
+(* NOTATION
+Notation RX := (cpoly_cring R).
+*)
+
+(*#*
+*** Constant and identity
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_X_.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_C_.con".
+
+(* UNEXPORTED
+Hint Resolve cpoly_X_ cpoly_C_: algebra.
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_const_eq.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_c_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_c_one.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_c_mult.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_lin.con".
+
+(* UNEXPORTED
+Hint Resolve cpoly_lin: algebra.
+*)
+
+(* SUPERFLUOUS *)
+
+inline "cic:/CoRN/algebra/CPolynomials/poly_linear.con".
+
+(* UNEXPORTED
+Hint Resolve _c_zero: algebra.
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/poly_c_apzero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_c_mult_lin.con".
+
+(* SUPERFLUOUS ? *)
+
+inline "cic:/CoRN/algebra/CPolynomials/lin_mult.con".
+
+(* UNEXPORTED
+Hint Resolve lin_mult: algebra.
+*)
+
+(*#* *** Application of polynomials
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/poly_eq_zero.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/apply_wd.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpolyap_pres_eq.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpolyap_strext.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_csetoid_op.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_c_apply.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_x_apply.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/plus_apply.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/inv_apply.con".
+
+(* UNEXPORTED
+Hint Resolve plus_apply inv_apply: algebra.
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/minus_apply.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/_c_mult_apply.con".
+
+(* UNEXPORTED
+Hint Resolve _c_mult_apply: algebra.
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/mult_apply.con".
+
+(* UNEXPORTED
+Hint Resolve mult_apply: algebra.
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/one_apply.con".
+
+(* UNEXPORTED
+Hint Resolve one_apply: algebra.
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/nexp_apply.con".
+
+(* SUPERFLUOUS *)
+
+inline "cic:/CoRN/algebra/CPolynomials/poly_inv_apply.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/Sum0_cpoly_ap.con".
+
+inline "cic:/CoRN/algebra/CPolynomials/Sum_cpoly_ap.con".
+
+(* UNEXPORTED
+End Poly_properties
+*)
+
+(*#* ** Induction properties of polynomials for [Prop]
+*)
+
+(* UNEXPORTED
+Section Poly_Prop_Induction
+*)
+
+alias id "CR" = "cic:/CoRN/algebra/CPolynomials/Poly_Prop_Induction/CR.var".
+
+(* NOTATION
+Notation Cpoly := (cpoly CR).
+*)
+
+(* NOTATION
+Notation Cpoly_zero := (cpoly_zero CR).
+*)
+
+(* NOTATION
+Notation Cpoly_linear := (cpoly_linear CR).
+*)
+
+(* NOTATION
+Notation Cpoly_cring := (cpoly_cring CR).
+*)
+
+inline "cic:/CoRN/algebra/CPolynomials/cpoly_double_ind.con".
+
+(* UNEXPORTED
+End Poly_Prop_Induction
+*)
+
+(* UNEXPORTED
+Hint Resolve poly_linear cpoly_lin: algebra.
+*)
+
+(* UNEXPORTED
+Hint Resolve apply_wd cpoly_const_eq: algebra_c.
+*)
+
+(* UNEXPORTED
+Hint Resolve _c_apply _x_apply inv_apply plus_apply minus_apply mult_apply
+  nexp_apply: algebra.
+*)
+
+(* UNEXPORTED
+Hint Resolve one_apply _c_zero _c_one _c_mult: algebra.
+*)
+
+(* UNEXPORTED
+Hint Resolve poly_inv_apply: algebra.
+*)
+
+(* UNEXPORTED
+Hint Resolve _c_mult_lin: algebra.
+*)
+