--- /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 *********************)
+
+set "baseuri" "cic:/matita/CoRN-Decl/reals/Max_AbsIR".
+
+include "CoRN.ma".
+
+(* $Id: Max_AbsIR.v,v 1.13 2004/04/23 10:01:04 lcf Exp $ *)
+
+(*#* printing Max %\ensuremath{\max}% *)
+
+(*#* printing Min %\ensuremath{\min}% *)
+
+include "reals/CauchySeq.ma".
+
+(* UNEXPORTED
+Section Maximum
+*)
+
+(* UNEXPORTED
+Section Max_function
+*)
+
+(*#* ** Maximum, Minimum and Absolute Value
+
+%\begin{convention}%
+Let [x] and [y] be reals
+(we will define the maximum of [x] and [y]).
+%\end{convention}%
+*)
+
+alias id "x" = "cic:/CoRN/reals/Max_AbsIR/Maximum/Max_function/x.var".
+
+alias id "y" = "cic:/CoRN/reals/Max_AbsIR/Maximum/Max_function/y.var".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Max_seq.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Max_seq_char.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Cauchy_Max_seq.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Max_CauchySeq.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/MAX.con".
+
+(*#*
+Constructively, the elementary properties of the maximum function are:
+- [x [<=] Max (x,y)],
+- [x [<=] Max (y,x)],
+- [z [<] Max(x,y) -> z [<] x or z [<] y].
+
+(This can be more concisely expressed as
+[z [<] Max(x,y) Iff z [<] x or z [<] y]).
+From these elementary properties we can prove all other properties, including
+strong extensionality.
+With strong extensionality, we can make the binary operation [Max].
+(So [Max] is [MAX] coupled with some proofs.)
+*)
+
+inline "cic:/CoRN/reals/Max_AbsIR/lft_leEq_MAX.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/rht_leEq_MAX.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/less_MAX_imp.con".
+
+(* UNEXPORTED
+End Max_function
+*)
+
+inline "cic:/CoRN/reals/Max_AbsIR/MAX_strext.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/MAX_wd.con".
+
+(* UNEXPORTED
+Section properties_of_Max
+*)
+
+(*#* *** Maximum *)
+
+inline "cic:/CoRN/reals/Max_AbsIR/Max.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Max_wd_unfolded.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/lft_leEq_Max.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/rht_leEq_Max.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/less_Max_imp.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Max_leEq.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Max_less.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/equiv_imp_eq_max.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Max_id.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Max_comm.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/leEq_imp_Max_is_rht.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Max_is_rht_imp_leEq.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Max_minus_eps_leEq.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/max_one_ap_zero.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/pos_max_one.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/x_div_Max_leEq_x.con".
+
+(* UNEXPORTED
+End properties_of_Max
+*)
+
+(* UNEXPORTED
+End Maximum
+*)
+
+(* UNEXPORTED
+Hint Resolve Max_id: algebra.
+*)
+
+(* UNEXPORTED
+Section Minimum
+*)
+
+(*#* *** Mininum
+
+The minimum is defined by the formula
+[Min(x,y) [=] [--]Max( [--]x,[--]y)].
+*)
+
+inline "cic:/CoRN/reals/Max_AbsIR/MIN.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/MIN_wd.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/MIN_strext.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min_wd_unfolded.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min_strext_unfolded.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min_leEq_lft.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min_leEq_rht.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min_less_imp.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/leEq_Min.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/less_Min.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/equiv_imp_eq_min.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min_id.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min_comm.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/leEq_imp_Min_is_lft.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min_is_lft_imp_leEq.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/leEq_Min_plus_eps.con".
+
+alias id "a" = "cic:/CoRN/reals/Max_AbsIR/Minimum/a.var".
+
+alias id "b" = "cic:/CoRN/reals/Max_AbsIR/Minimum/b.var".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min_leEq_Max.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min_leEq_Max'.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min3_leEq_Max3.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min_less_Max.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/ap_imp_Min_less_Max.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Min_less_Max_imp_ap.con".
+
+(* UNEXPORTED
+End Minimum
+*)
+
+(*#**********************************)
+
+(* UNEXPORTED
+Section Absolute
+*)
+
+(*#**********************************)
+
+(*#* *** Absolute value *)
+
+inline "cic:/CoRN/reals/Max_AbsIR/ABSIR.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/ABSIR_strext.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/ABSIR_wd.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_wd.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_wdl.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_wdr.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIRz_isz.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_nonneg.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_pos.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_cancel_ap_zero.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_resp_ap_zero.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/leEq_AbsIR.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/inv_leEq_AbsIR.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsSmall_e.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsSmall_imp_AbsIR.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_eq_AbsSmall.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_imp_AbsSmall.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsSmall_transitive.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/zero_less_AbsIR_plus_one.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_inv.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_minus.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_eq_x.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_eq_inv_x.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/less_AbsIR.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/leEq_distr_AbsIR.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_approach_zero.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_eq_zero.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Abs_Max.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_str_bnd.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/AbsIR_bnd.con".
+
+(* UNEXPORTED
+End Absolute
+*)
+
+(* UNEXPORTED
+Hint Resolve AbsIRz_isz: algebra.
+*)
+
+(* UNEXPORTED
+Section Part_Function_Max
+*)
+
+(*#* *** Functional Operators
+
+The existence of these operators allows us to lift them to functions. We will define the maximum, minimum and absolute value of two partial functions.
+
+%\begin{convention}%
+Let [F,G:PartIR] and denote by [P] and [Q] their respective domains.
+%\end{convention}%
+*)
+
+alias id "F" = "cic:/CoRN/reals/Max_AbsIR/Part_Function_Max/F.var".
+
+alias id "G" = "cic:/CoRN/reals/Max_AbsIR/Part_Function_Max/G.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/reals/Max_AbsIR/Part_Function_Max/P.con" "Part_Function_Max__".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Part_Function_Max/Q.con" "Part_Function_Max__".
+
+(* end hide *)
+
+inline "cic:/CoRN/reals/Max_AbsIR/part_function_Max_strext.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/FMax.con".
+
+(* UNEXPORTED
+End Part_Function_Max
+*)
+
+(* UNEXPORTED
+Section Part_Function_Abs
+*)
+
+alias id "F" = "cic:/CoRN/reals/Max_AbsIR/Part_Function_Abs/F.var".
+
+alias id "G" = "cic:/CoRN/reals/Max_AbsIR/Part_Function_Abs/G.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/reals/Max_AbsIR/Part_Function_Abs/P.con" "Part_Function_Abs__".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Part_Function_Abs/Q.con" "Part_Function_Abs__".
+
+(* end hide *)
+
+inline "cic:/CoRN/reals/Max_AbsIR/FMin.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/FAbs.con".
+
+(* UNEXPORTED
+Opaque Max.
+*)
+
+inline "cic:/CoRN/reals/Max_AbsIR/FMin_char.con".
+
+(* UNEXPORTED
+Transparent Max.
+*)
+
+inline "cic:/CoRN/reals/Max_AbsIR/FAbs_char.con".
+
+(* UNEXPORTED
+End Part_Function_Abs
+*)
+
+(* UNEXPORTED
+Hint Resolve FAbs_char: algebra.
+*)
+
+inline "cic:/CoRN/reals/Max_AbsIR/FAbs_char'.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/FAbs_nonneg.con".
+
+(* UNEXPORTED
+Hint Resolve FAbs_char': algebra.
+*)
+
+(* UNEXPORTED
+Section Inclusion
+*)
+
+alias id "F" = "cic:/CoRN/reals/Max_AbsIR/Inclusion/F.var".
+
+alias id "G" = "cic:/CoRN/reals/Max_AbsIR/Inclusion/G.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/reals/Max_AbsIR/Inclusion/P.con" "Inclusion__".
+
+inline "cic:/CoRN/reals/Max_AbsIR/Inclusion/Q.con" "Inclusion__".
+
+(* end hide *)
+
+(*#*
+%\begin{convention}% Let [R:IR->CProp].
+%\end{convention}%
+*)
+
+alias id "R" = "cic:/CoRN/reals/Max_AbsIR/Inclusion/R.var".
+
+inline "cic:/CoRN/reals/Max_AbsIR/included_FMax.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/included_FMax'.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/included_FMax''.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/included_FMin.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/included_FMin'.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/included_FMin''.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/included_FAbs.con".
+
+inline "cic:/CoRN/reals/Max_AbsIR/included_FAbs'.con".
+
+(* UNEXPORTED
+End Inclusion
+*)
+
+(* UNEXPORTED
+Hint Resolve included_FMax included_FMin included_FAbs : included.
+*)
+
+(* UNEXPORTED
+Hint Immediate included_FMax' included_FMin' included_FAbs'
+ included_FMax'' included_FMin'' : included.
+*)
+
projects / helm.git / blobdiff