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stdpp
Commits
cbb67c16
Commit
cbb67c16
authored
Jan 29, 2019
by
Robbert Krebbers
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Make rtc an instance of PreOrder.
parent
9883e152
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#14190
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in 10 minutes and 32 seconds
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theories/relations.v
theories/relations.v
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theories/relations.v
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cbb67c16
...
...
@@ 65,19 +65,20 @@ Section rtc.
Hint
Constructors
rtc
nsteps
bsteps
tc
:
core
.
Lemma
rtc_transitive
x
y
z
:
rtc
R
x
y
→
rtc
R
y
z
→
rtc
R
x
z
.
Proof
.
induction
1
;
eauto
.
Qed
.
(* We give this instance a lowerthanusual priority because [setoid_rewrite]
queries for [@Reflexive Prop ?r] in the hope of [iff_reflexive] getting
picked as the instance. [rtc_reflexive] overlaps with that, leading to
backtracking. We cannot set [Hint Mode] because that query must not fail,
but we can at least avoid picking [rtc_reflexive].
See Coq bug https://github.com/coq/coq/issues/7916. *)
Global
Instance
rtc_reflexive
:
Reflexive
(
rtc
R
)

10
.
Proof
.
exact
(@
rtc_refl
A
R
).
Qed
.
Lemma
rtc_transitive
x
y
z
:
rtc
R
x
y
→
rtc
R
y
z
→
rtc
R
x
z
.
Proof
.
induction
1
;
eauto
.
Qed
.
Global
Instance
:
Transitive
(
rtc
R
).
Proof
.
exact
rtc_transitive
.
Qed
.
See Coq bug https://github.com/coq/coq/issues/7916 and the test
[tests.typeclasses.test_setoid_rewrite]. *)
Global
Instance
rtc_po
:
PreOrder
(
rtc
R
)

10
.
Proof
.
split
.
exact
(@
rtc_refl
A
R
).
exact
rtc_transitive
.
Qed
.
Lemma
rtc_once
x
y
:
R
x
y
→
rtc
R
x
y
.
Proof
.
eauto
.
Qed
.
Lemma
rtc_r
x
y
z
:
rtc
R
x
y
→
R
y
z
→
rtc
R
x
z
.
...
...
@@ 160,7 +161,7 @@ Section rtc.
Lemma
tc_transitive
x
y
z
:
tc
R
x
y
→
tc
R
y
z
→
tc
R
x
z
.
Proof
.
induction
1
;
eauto
.
Qed
.
Global
Instance
:
Transitive
(
tc
R
).
Global
Instance
tc_transitive'
:
Transitive
(
tc
R
).
Proof
.
exact
tc_transitive
.
Qed
.
Lemma
tc_r
x
y
z
:
tc
R
x
y
→
R
y
z
→
tc
R
x
z
.
Proof
.
intros
.
etrans
;
eauto
.
Qed
.
...
...
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