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Simon Spies
stdpp
Commits
3db8e2f7
Commit
3db8e2f7
authored
Jun 30, 2019
by
Ralf Jung
Committed by
Robbert
Jun 30, 2019
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add invserses of bool_decide_{true,false}
parent
3b8eed3c
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theories/decidable.v
theories/decidable.v
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theories/decidable.v
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3db8e2f7
...
@@ -112,14 +112,29 @@ Proof. rewrite bool_decide_spec; trivial. Qed.
...
@@ -112,14 +112,29 @@ Proof. rewrite bool_decide_spec; trivial. Qed.
Lemma
bool_decide_pack
(
P
:
Prop
)
{
dec
:
Decision
P
}
:
P
→
bool_decide
P
.
Lemma
bool_decide_pack
(
P
:
Prop
)
{
dec
:
Decision
P
}
:
P
→
bool_decide
P
.
Proof
.
rewrite
bool_decide_spec
;
trivial
.
Qed
.
Proof
.
rewrite
bool_decide_spec
;
trivial
.
Qed
.
Hint
Resolve
bool_decide_pack
:
core
.
Hint
Resolve
bool_decide_pack
:
core
.
Lemma
bool_decide_true
(
P
:
Prop
)
`
{
Decision
P
}
:
P
→
bool_decide
P
=
true
.
Proof
.
case_bool_decide
;
tauto
.
Qed
.
Lemma
bool_decide_eq_true
(
P
:
Prop
)
`
{
Decision
P
}
:
bool_decide
P
=
true
↔
P
.
Lemma
bool_decide_false
(
P
:
Prop
)
`
{
Decision
P
}
:
¬
P
→
bool_decide
P
=
false
.
Proof
.
case_bool_decide
;
intuition
discriminate
.
Qed
.
Proof
.
case_bool_decide
;
tauto
.
Qed
.
Lemma
bool_decide_eq_false
(
P
:
Prop
)
`
{
Decision
P
}
:
bool_decide
P
=
false
↔
¬
P
.
Proof
.
case_bool_decide
;
intuition
discriminate
.
Qed
.
Lemma
bool_decide_iff
(
P
Q
:
Prop
)
`
{
Decision
P
,
Decision
Q
}
:
Lemma
bool_decide_iff
(
P
Q
:
Prop
)
`
{
Decision
P
,
Decision
Q
}
:
(
P
↔
Q
)
→
bool_decide
P
=
bool_decide
Q
.
(
P
↔
Q
)
→
bool_decide
P
=
bool_decide
Q
.
Proof
.
repeat
case_bool_decide
;
tauto
.
Qed
.
Proof
.
repeat
case_bool_decide
;
tauto
.
Qed
.
Lemma
bool_decide_eq_true_1
P
`
{!
Decision
P
}
:
bool_decide
P
=
true
→
P
.
Proof
.
apply
bool_decide_eq_true
.
Qed
.
Lemma
bool_decide_eq_true_2
P
`
{!
Decision
P
}
:
P
→
bool_decide
P
=
true
.
Proof
.
apply
bool_decide_eq_true
.
Qed
.
Lemma
bool_decide_eq_false_1
P
`
{!
Decision
P
}
:
bool_decide
P
=
false
→
¬
P
.
Proof
.
apply
bool_decide_eq_false
.
Qed
.
Lemma
bool_decide_eq_false_2
P
`
{!
Decision
P
}
:
¬
P
→
bool_decide
P
=
false
.
Proof
.
apply
bool_decide_eq_false
.
Qed
.
(** Backwards compatibility notations. *)
Notation
bool_decide_true
:
=
bool_decide_eq_true_2
.
Notation
bool_decide_false
:
=
bool_decide_eq_false_2
.
(** * Decidable Sigma types *)
(** * Decidable Sigma types *)
(** Leibniz equality on Sigma types requires the equipped proofs to be
(** Leibniz equality on Sigma types requires the equipped proofs to be
equal as Coq does not support proof irrelevance. For decidable we
equal as Coq does not support proof irrelevance. For decidable we
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