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Rice Wine
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b8dc0773
Commit
b8dc0773
authored
Mar 10, 2017
by
Ralf Jung
Browse files
make frac_included into general lemma about < and + of positive fractions
parent
26e93ebf
Changes
1
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Inline
Side-by-side
theories/algebra/frac.v
View file @
b8dc0773
...
...
@@ -11,14 +11,19 @@ Instance frac_valid : Valid frac := λ x, (x ≤ 1)%Qc.
Instance
frac_pcore
:
PCore
frac
:
=
λ
_
,
None
.
Instance
frac_op
:
Op
frac
:
=
λ
x
y
,
(
x
+
y
)%
Qp
.
Lemma
frac_included
(
x
y
:
frac
)
:
x
≼
y
↔
(
x
<
y
)%
Qc
.
(* TODO: Find better place for this lemma. *)
Lemma
Qp_le_sum
(
x
y
:
Qp
)
:
(
x
<
y
)%
Qc
↔
(
∃
z
,
y
=
x
+
z
)%
Qp
.
Proof
.
split
.
-
intros
[
z
->%
leibniz_equiv
]
;
simpl
.
rewrite
-{
1
}(
Qcplus_0_r
x
).
apply
Qcplus_lt_mono_l
,
Qp_prf
.
-
intros
Hlt
%
Qclt_minus_iff
.
exists
(
mk_Qp
(
y
-
x
)
Hlt
).
apply
Qp_eq
;
simpl
.
by
rewrite
(
Qcplus_comm
y
)
Qcplus_assoc
Qcplus_opp_r
Qcplus_0_l
.
-
intros
[
z
->%
leibniz_equiv
]
;
simpl
.
rewrite
-{
1
}(
Qcplus_0_r
x
).
apply
Qcplus_lt_mono_l
,
Qp_prf
.
Qed
.
Lemma
frac_included
(
x
y
:
frac
)
:
x
≼
y
↔
(
x
<
y
)%
Qc
.
Proof
.
symmetry
.
exact
:
Qp_le_sum
.
Qed
.
Corollary
frac_included_weak
(
x
y
:
frac
)
:
x
≼
y
→
(
x
≤
y
)%
Qc
.
Proof
.
intros
?%
frac_included
.
auto
using
Qclt_le_weak
.
Qed
.
...
...
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