| Axiomatisation of the classical reals |
Require Export ZArith_base.
Require Export Rdefinitions.
Open Local Scope R_scope.
| Addition |
Axiom Rplus_comm : forall r1 r2:R, r1 + r2 = r2 + r1.
Hint Resolve Rplus_comm: real.
Axiom Rplus_assoc : forall r1 r2 r3:R, r1 + r2 + r3 = r1 + (r2 + r3).
Hint Resolve Rplus_assoc: real.
Axiom Rplus_opp_r : forall r:R, r + - r = 0.
Hint Resolve Rplus_opp_r: real v62.
Axiom Rplus_0_l : forall r:R, 0 + r = r.
Hint Resolve Rplus_0_l: real.
| Multiplication |
Axiom Rmult_comm : forall r1 r2:R, r1 * r2 = r2 * r1.
Hint Resolve Rmult_comm: real v62.
Axiom Rmult_assoc : forall r1 r2 r3:R, r1 * r2 * r3 = r1 * (r2 * r3).
Hint Resolve Rmult_assoc: real v62.
Axiom Rinv_l : forall r:R, r <> 0 -> / r * r = 1.
Hint Resolve Rinv_l: real.
Axiom Rmult_1_l : forall r:R, 1 * r = r.
Hint Resolve Rmult_1_l: real.
Axiom R1_neq_R0 : 1 <> 0.
Hint Resolve R1_neq_R0: real.
| Distributivity |
Axiom
Rmult_plus_distr_l : forall r1 r2 r3:R, r1 * (r2 + r3) = r1 * r2 + r1 * r3.
Hint Resolve Rmult_plus_distr_l: real v62.
| Order axioms |
| Total Order |
Axiom total_order_T : forall r1 r2:R, {r1 < r2} + {r1 = r2} + {r1 > r2}.
| Lower |
Axiom Rlt_asym : forall r1 r2:R, r1 < r2 -> ~ r2 < r1.
Axiom Rlt_trans : forall r1 r2 r3:R, r1 < r2 -> r2 < r3 -> r1 < r3.
Axiom Rplus_lt_compat_l : forall r r1 r2:R, r1 < r2 -> r + r1 < r + r2.
Axiom
Rmult_lt_compat_l : forall r r1 r2:R, 0 < r -> r1 < r2 -> r * r1 < r * r2.
Hint Resolve Rlt_asym Rplus_lt_compat_l Rmult_lt_compat_l: real.
| Injection from N to R |
Fixpoint INR (n:nat) : R :=
match n with
| O => 0
| S O => 1
| S n => INR n + 1
end.
Arguments Scope INR [nat_scope].
Injection from Z to R
|
Definition IZR (z:Z) : R :=
match z with
| Z0 => 0
| Zpos n => INR (nat_of_P n)
| Zneg n => - INR (nat_of_P n)
end.
Arguments Scope IZR [Z_scope].
R Archimedian
|
Axiom archimed : forall r:R, IZR (up r) > r /\ IZR (up r) - r <= 1.
R Complete
|
Definition is_upper_bound (E:R -> Prop) (m:R) := forall x:R, E x -> x <= m.
Definition bound (E:R -> Prop) := exists m : R, is_upper_bound E m.
Definition is_lub (E:R -> Prop) (m:R) :=
is_upper_bound E m /\ (forall b:R, is_upper_bound E b -> m <= b).
Axiom
completeness :
forall E:R -> Prop,
bound E -> (exists x : R, E x) -> sigT (fun m:R => is_lub E m).