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(*i $Id$ i*)
(*********************************************************)
(*s {Axiomatisation of the classical reals} *)
(*********************************************************)
Require Export ZArith.
Require Export Rsyntax.
Require Export TypeSyntax.
(*********************************************************)
(* {Field axioms} *)
(*********************************************************)
(*********************************************************)
(*s Addition *)
(*********************************************************)
(**********)
Axiom Rplus_sym:(r1,r2:R)``r1+r2==r2+r1``.
Hints Resolve Rplus_sym : real.
(**********)
Axiom Rplus_assoc:(r1,r2,r3:R)``(r1+r2)+r3==r1+(r2+r3)``.
Hints Resolve Rplus_assoc : real.
(**********)
Axiom Rplus_Ropp_r:(r:R)``r+(-r)==0``.
Hints Resolve Rplus_Ropp_r : real v62.
(**********)
Axiom Rplus_Ol:(r:R)``0+r==r``.
Hints Resolve Rplus_Ol : real.
(***********************************************************)
(*s Multiplication *)
(***********************************************************)
(**********)
Axiom Rmult_sym:(r1,r2:R)``r1*r2==r2*r1``.
Hints Resolve Rmult_sym : real v62.
(**********)
Axiom Rmult_assoc:(r1,r2,r3:R)``(r1*r2)*r3==r1*(r2*r3)``.
Hints Resolve Rmult_assoc : real v62.
(**********)
Axiom Rinv_l:(r:R)``r<>0``->``(1/r)*r==1``.
Hints Resolve Rinv_l : real.
(**********)
Axiom Rmult_1l:(r:R)``1*r==r``.
Hints Resolve Rmult_1l : real.
(**********)
Axiom R1_neq_R0:``1<>0``.
Hints Resolve R1_neq_R0 : real.
(*********************************************************)
(*s Distributivity *)
(*********************************************************)
(**********)
Axiom Rmult_Rplus_distr:(r1,r2,r3:R)``r1*(r2+r3)==(r1*r2)+(r1*r3)``.
Hints Resolve Rmult_Rplus_distr : real v62.
(*********************************************************)
(* Order axioms *)
(*********************************************************)
(*********************************************************)
(*s Total Order *)
(*********************************************************)
(**********)
Axiom total_order_T:(r1,r2:R)(sumorT (sumboolT ``r1<r2`` r1==r2) ``r1>r2``).
(*********************************************************)
(*s Lower *)
(*********************************************************)
(**********)
Axiom Rlt_antisym:(r1,r2:R)``r1<r2`` -> ~ ``r2<r1``.
(**********)
Axiom Rlt_trans:(r1,r2,r3:R)
``r1<r2``->``r2<r3``->``r1<r3``.
(**********)
Axiom Rlt_compatibility:(r,r1,r2:R)``r1<r2``->``r+r1<r+r2``.
(**********)
Axiom Rlt_monotony:(r,r1,r2:R)``0<r``->``r1<r2``->``r*r1<r*r2``.
Hints Resolve Rlt_antisym Rlt_compatibility Rlt_monotony : real.
(**********************************************************)
(*s Injection from N to R *)
(**********************************************************)
(**********)
Fixpoint INR [n:nat]:R:=(Cases n of
O => ``0``
|(S O) => ``1``
|(S n) => ``(INR n)+1``
end).
(**********************************************************)
(*s Injection from Z to R *)
(**********************************************************)
(**********)
Definition IZR:Z->R:=[z:Z](Cases z of
ZERO => ``0``
|(POS n) => (INR (convert n))
|(NEG n) => ``-(INR (convert n))``
end).
(**********************************************************)
(*s R Archimedian *)
(**********************************************************)
(**********)
Axiom archimed:(r:R)``(IZR (up r)) > r``/\``(IZR (up r))-r <= 1``.
(**********************************************************)
(*s R Complete *)
(**********************************************************)
(**********)
Definition is_upper_bound:=[E:R->Prop][m:R](x:R)(E x)->``x <= m``.
(**********)
Definition bound:=[E:R->Prop](ExT [m:R](is_upper_bound E m)).
(**********)
Definition is_lub:=[E:R->Prop][m:R]
(is_upper_bound E m)/\(b:R)(is_upper_bound E b)->``m <= b``.
(**********)
Axiom complet:(E:R->Prop)(bound E)->
(ExT [x:R] (E x))->
(ExT [m:R](is_lub E m)).
|
