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stdpp
Commits
3db8e2f7
Commit
3db8e2f7
authored
5 years ago
by
Ralf Jung
Committed by
Robbert Krebbers
5 years ago
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add invserses of bool_decide_{true,false}
parent
3b8eed3c
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theories/decidable.v
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theories/decidable.v
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4 deletions
theories/decidable.v
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3db8e2f7
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@@ -112,14 +112,29 @@ Proof. rewrite bool_decide_spec; trivial. Qed.
Lemma
bool_decide_pack
(
P
:
Prop
)
{
dec
:
Decision
P
}
:
P
→
bool_decide
P
.
Proof
.
rewrite
bool_decide_spec
;
trivial
.
Qed
.
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_false
(
P
:
Prop
)
`{
Decision
P
}
:
¬
P
→
bool_decide
P
=
false
.
Proof
.
case_bool_decide
;
tauto
.
Qed
.
Lemma
bool_decide_eq_true
(
P
:
Prop
)
`{
Decision
P
}
:
bool_decide
P
=
true
↔
P
.
Proof
.
case_bool_decide
;
intuition
discriminate
.
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
}
:
(
P
↔
Q
)
→
bool_decide
P
=
bool_decide
Q
.
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 *)
(** Leibniz equality on Sigma types requires the equipped proofs to be
equal as Coq does not support proof irrelevance. For decidable we
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