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George Pirlea
Iris
Commits
39c7307f
Commit
39c7307f
authored
Mar 03, 2016
by
Robbert Krebbers
Browse files
Make naive_solver deal with some Boolean connectives.
parent
8ca359a5
Changes
9
Hide whitespace changes
Inline
Sidebyside
prelude/decidable.v
View file @
39c7307f
...
...
@@ 11,7 +11,7 @@ Lemma dec_stable `{Decision P} : ¬¬P → P.
Proof
.
firstorder
.
Qed
.
Lemma
Is_true_reflect
(
b
:
bool
)
:
reflect
b
b
.
Proof
.
destruct
b
.
by
left
.
right
.
intros
[].
Qed
.
Proof
.
destruct
b
.
left
;
constructor
.
right
.
intros
[].
Qed
.
Instance
:
Inj
(=)
(
↔
)
Is_true
.
Proof
.
intros
[]
[]
;
simpl
;
intuition
.
Qed
.
...
...
@@ 24,17 +24,17 @@ Definition decide_rel {A B} (R : A → B → Prop) {dec : ∀ x y, Decision (R x
(
x
:
A
)
(
y
:
B
)
:
Decision
(
R
x
y
)
:
=
dec
x
y
.
Lemma
decide_rel_correct
{
A
B
}
(
R
:
A
→
B
→
Prop
)
`
{
∀
x
y
,
Decision
(
R
x
y
)}
(
x
:
A
)
(
y
:
B
)
:
decide_rel
R
x
y
=
decide
(
R
x
y
).
Proof
.
done
.
Qed
.
Proof
.
reflexivity
.
Qed
.
Lemma
decide_True
{
A
}
`
{
Decision
P
}
(
x
y
:
A
)
:
P
→
(
if
decide
P
then
x
else
y
)
=
x
.
Proof
.
by
destruct
(
decide
P
).
Qed
.
Proof
.
destruct
(
decide
P
)
;
tauto
.
Qed
.
Lemma
decide_False
{
A
}
`
{
Decision
P
}
(
x
y
:
A
)
:
¬
P
→
(
if
decide
P
then
x
else
y
)
=
y
.
Proof
.
by
destruct
(
decide
P
).
Qed
.
Proof
.
destruct
(
decide
P
)
;
tauto
.
Qed
.
Lemma
decide_iff
{
A
}
P
Q
`
{
Decision
P
,
Decision
Q
}
(
x
y
:
A
)
:
(
P
↔
Q
)
→
(
if
decide
P
then
x
else
y
)
=
(
if
decide
Q
then
x
else
y
).
Proof
.
intros
[??].
destruct
(
decide
P
),
(
decide
Q
)
;
intuition
.
Qed
.
Proof
.
intros
[??].
destruct
(
decide
P
),
(
decide
Q
)
;
tauto
.
Qed
.
(** The tactic [destruct_decide] destructs a sumbool [dec]. If one of the
components is double negated, it will try to remove the double negation. *)
...
...
@@ 95,7 +95,7 @@ Definition bool_decide (P : Prop) {dec : Decision P} : bool :=
if
dec
then
true
else
false
.
Lemma
bool_decide_reflect
P
`
{
dec
:
Decision
P
}
:
reflect
P
(
bool_decide
P
).
Proof
.
unfold
bool_decide
.
destruct
dec
.
by
left
.
by
right
.
Qed
.
Proof
.
unfold
bool_decide
.
destruct
dec
;
[
left

right
]
;
assumption
.
Qed
.
Tactic
Notation
"case_bool_decide"
"as"
ident
(
Hd
)
:
=
match
goal
with
...
...
@@ 108,15 +108,15 @@ Tactic Notation "case_bool_decide" :=
let
H
:
=
fresh
in
case_bool_decide
as
H
.
Lemma
bool_decide_spec
(
P
:
Prop
)
{
dec
:
Decision
P
}
:
bool_decide
P
↔
P
.
Proof
.
unfold
bool_decide
.
by
destruct
dec
.
Qed
.
Proof
.
unfold
bool_decide
.
destruct
dec
;
simpl
;
tauto
.
Qed
.
Lemma
bool_decide_unpack
(
P
:
Prop
)
{
dec
:
Decision
P
}
:
bool_decide
P
→
P
.
Proof
.
by
rewrite
bool_decide_spec
.
Qed
.
Proof
.
rewrite
bool_decide_spec
;
trivial
.
Qed
.
Lemma
bool_decide_pack
(
P
:
Prop
)
{
dec
:
Decision
P
}
:
P
→
bool_decide
P
.
Proof
.
by
rewrite
bool_decide_spec
.
Qed
.
Proof
.
rewrite
bool_decide_spec
;
trivial
.
Qed
.
Lemma
bool_decide_true
(
P
:
Prop
)
`
{
Decision
P
}
:
P
→
bool_decide
P
=
true
.
Proof
.
by
case_bool_decide
.
Qed
.
Proof
.
case_bool_decide
;
tauto
.
Qed
.
Lemma
bool_decide_false
(
P
:
Prop
)
`
{
Decision
P
}
:
¬
P
→
bool_decide
P
=
false
.
Proof
.
by
case_bool_decide
.
Qed
.
Proof
.
case_bool_decide
;
tauto
.
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
.
...
...
@@ 138,7 +138,7 @@ Lemma dsig_eq `(P : A → Prop) `{∀ x, Decision (P x)}
Proof
.
apply
(
sig_eq_pi
_
).
Qed
.
Lemma
dexists_proj1
`
(
P
:
A
→
Prop
)
`
{
∀
x
,
Decision
(
P
x
)}
(
x
:
dsig
P
)
p
:
dexist
(
`
x
)
p
=
x
.
Proof
.
by
apply
dsig_eq
.
Qed
.
Proof
.
apply
dsig_eq
;
reflexivity
.
Qed
.
(** * Instances of Decision *)
(** Instances of [Decision] for operators of propositional logic. *)
...
...
@@ 184,7 +184,7 @@ Instance uncurry_dec `(P_dec : ∀ (p : A * B), Decision (P p)) x y :
Instance
sig_eq_dec
`
(
P
:
A
→
Prop
)
`
{
∀
x
,
ProofIrrel
(
P
x
)}
`
{
∀
x
y
:
A
,
Decision
(
x
=
y
)}
(
x
y
:
sig
P
)
:
Decision
(
x
=
y
).
Proof
.
refine
(
cast_if
(
decide
(
`
x
=
`
y
)))
;
by
rewrite
sig_eq_pi
.
Defined
.
Proof
.
refine
(
cast_if
(
decide
(
`
x
=
`
y
)))
;
rewrite
sig_eq_pi
;
trivial
.
Defined
.
(** Some laws for decidable propositions *)
Lemma
not_and_l
{
P
Q
:
Prop
}
`
{
Decision
P
}
:
¬
(
P
∧
Q
)
↔
¬
P
∨
¬
Q
.
...
...
prelude/list.v
View file @
39c7307f
...
...
@@ 3,7 +3,7 @@
(** This file collects general purpose definitions and theorems on lists that
are not in the Coq standard library. *)
From
Coq
Require
Export
Permutation
.
From
prelude
Require
Export
numbers
base
decidable
option
.
From
prelude
Require
Export
numbers
base
option
.
Arguments
length
{
_
}
_
.
Arguments
cons
{
_
}
_
_
.
...
...
prelude/listset.v
View file @
39c7307f
...
...
@@ 2,7 +2,7 @@
(* This file is distributed under the terms of the BSD license. *)
(** This file implements finite set as unordered lists without duplicates
removed. This implementation forms a monad. *)
From
prelude
Require
Export
base
decidable
collections
list
.
From
prelude
Require
Export
collections
list
.
Record
listset
A
:
=
Listset
{
listset_car
:
list
A
}.
Arguments
listset_car
{
_
}
_
.
...
...
prelude/listset_nodup.v
View file @
39c7307f
...
...
@@ 3,7 +3,7 @@
(** This file implements finite as unordered lists without duplicates.
Although this implementation is slow, it is very useful as decidable equality
is the only constraint on the carrier set. *)
From
prelude
Require
Export
base
decidable
collections
list
.
From
prelude
Require
Export
collections
list
.
Record
listset_nodup
A
:
=
ListsetNoDup
{
listset_nodup_car
:
list
A
;
listset_nodup_prf
:
NoDup
listset_nodup_car
...
...
prelude/option.v
View file @
39c7307f
...
...
@@ 2,7 +2,7 @@
(* This file is distributed under the terms of the BSD license. *)
(** This file collects general purpose definitions and theorems on the option
data type that are not in the Coq standard library. *)
From
prelude
Require
Export
base
tactics
decidable
.
From
prelude
Require
Export
tactics
.
Inductive
option_reflect
{
A
}
(
P
:
A
→
Prop
)
(
Q
:
Prop
)
:
option
A
→
Type
:
=

ReflectSome
x
:
P
x
→
option_reflect
P
Q
(
Some
x
)
...
...
prelude/orders.v
View file @
39c7307f
...
...
@@ 3,7 +3,7 @@
(** This file collects common properties of preorders and semi lattices. This
theory will mainly be used for the theory on collections and finite maps. *)
From
Coq
Require
Export
Sorted
.
From
prelude
Require
Export
base
decidable
tactics
list
.
From
prelude
Require
Export
tactics
list
.
(** * Arbitrary pre, parial and total orders *)
(** Properties about arbitrary pre, partial, and total orders. We do not use
...
...
prelude/prelude.v
View file @
39c7307f
...
...
@@ 3,7 +3,6 @@
From
prelude
Require
Export
base
tactics
decidable
orders
option
vector
...
...
prelude/proof_irrel.v
View file @
39c7307f
...
...
@@ 2,20 +2,20 @@
(* This file is distributed under the terms of the BSD license. *)
(** This file collects facts on proof irrelevant types/propositions. *)
From
Coq
Require
Import
Eqdep_dec
.
From
prelude
Require
Export
tactics
.
From
prelude
Require
Export
base
.
Hint
Extern
200
(
ProofIrrel
_
)
=>
progress
(
lazy
beta
)
:
typeclass_instances
.
Instance
:
ProofIrrel
True
.
Proof
.
by
intros
[]
[].
Qed
.
Proof
.
intros
[]
[]
;
reflexivity
.
Qed
.
Instance
:
ProofIrrel
False
.
Proof
.
by
intros
[].
Qed
.
Proof
.
intros
[].
Qed
.
Instance
and_pi
(
A
B
:
Prop
)
:
ProofIrrel
A
→
ProofIrrel
B
→
ProofIrrel
(
A
∧
B
).
Proof
.
intros
??
[??]
[??].
by
f_equal
.
Qed
.
Proof
.
intros
??
[??]
[??].
f_equal
;
trivial
.
Qed
.
Instance
prod_pi
(
A
B
:
Type
)
:
ProofIrrel
A
→
ProofIrrel
B
→
ProofIrrel
(
A
*
B
).
Proof
.
intros
??
[??]
[??].
by
f_equal
.
Qed
.
Proof
.
intros
??
[??]
[??].
f_equal
;
trivial
.
Qed
.
Instance
eq_pi
{
A
}
`
{
∀
x
y
:
A
,
Decision
(
x
=
y
)}
(
x
y
:
A
)
:
ProofIrrel
(
x
=
y
).
Proof
.
...
...
@@ 27,10 +27,10 @@ Proof. destruct b; simpl; apply _. Qed.
Lemma
sig_eq_pi
`
(
P
:
A
→
Prop
)
`
{
∀
x
,
ProofIrrel
(
P
x
)}
(
x
y
:
sig
P
)
:
x
=
y
↔
`
x
=
`
y
.
Proof
.
split
;
[
by
intros
<
].
split
;
[
intros
<
;
reflexivity
].
destruct
x
as
[
x
Hx
],
y
as
[
y
Hy
]
;
simpl
;
intros
;
subst
.
f_equal
.
apply
proof_irrel
.
Qed
.
Lemma
exists_proj1_pi
`
(
P
:
A
→
Prop
)
`
{
∀
x
,
ProofIrrel
(
P
x
)}
(
x
:
sig
P
)
p
:
`
x
↾
p
=
x
.
Proof
.
by
apply
(
sig_eq_pi
_
).
Qed
.
Proof
.
apply
(
sig_eq_pi
_
)
;
reflexivity
.
Qed
.
prelude/tactics.v
View file @
39c7307f
...
...
@@ 4,7 +4,7 @@
the development. *)
From
Coq
Require
Import
Omega
.
From
Coq
Require
Export
Psatz
.
From
prelude
Require
Export
bas
e
.
From
prelude
Require
Export
decidabl
e
.
Lemma
f_equal_dep
{
A
B
}
(
f
g
:
∀
x
:
A
,
B
x
)
x
:
f
=
g
→
f
x
=
g
x
.
Proof
.
intros
>
;
reflexivity
.
Qed
.
...
...
@@ 430,6 +430,8 @@ Tactic Notation "naive_solver" tactic(tac) :=

H
:
_
∧
_

_
=>
destruct
H

H
:
∃
_
,
_

_
=>
destruct
H

H
:
?P
→
?Q
,
H2
:
?P

_
=>
specialize
(
H
H2
)

H
:
Is_true
(
bool_decide
_
)

_
=>
apply
(
bool_decide_unpack
_
)
in
H

H
:
Is_true
(
_
&&
_
)

_
=>
apply
andb_True
in
H
;
destruct
H
(**i simplify and solve equalities *)


_
=>
progress
simplify_eq
/=
(**i solve the goal *)
...
...
@@ 441,6 +443,8 @@ Tactic Notation "naive_solver" tactic(tac) :=

reflexivity
]
(**i operations that generate more subgoals *)


_
∧
_
=>
split


Is_true
(
bool_decide
_
)
=>
apply
(
bool_decide_pack
_
)


Is_true
(
_
&&
_
)
=>
apply
andb_True
;
split

H
:
_
∨
_

_
=>
destruct
H
(**i solve the goal using the user supplied tactic *)


_
=>
solve
[
tac
]
...
...
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