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c490d858
Commit
c490d858
authored
Mar 14, 2013
by
Robbert Krebbers
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An axiomatization and implementation of machine integers.
parent
c7fe8cd2
Changes
5
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5 changed files
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95 additions
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21 deletions
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21
theories/base.v
theories/base.v
+6
0
theories/decidable.v
theories/decidable.v
+23
21
theories/list.v
theories/list.v
+11
0
theories/numbers.v
theories/numbers.v
+14
0
theories/proof_irrel.v
theories/proof_irrel.v
+41
0
No files found.
theories/base.v
View file @
c490d858
...
...
@@ 120,6 +120,12 @@ Instance sum_inhabited_r {A B} (iB : Inhabited A) : Inhabited (A + B) :=
end
.
Instance
option_inhabited
{
A
}
:
Inhabited
(
option
A
)
:
=
populate
None
.
(** ** Proof irrelevant types *)
(** This type class collects types that are proof irrelevant. That means, all
elements of the type are equal. We use this notion only used for propositions,
but by universe polymorphism we can generalize it. *)
Class
ProofIrrel
(
A
:
Type
)
:
Prop
:
=
proof_irrel
(
x
y
:
A
)
:
x
=
y
.
(** ** Setoid equality *)
(** We define an operational type class for setoid equality. This is based on
(Spitters/van der Weegen, 2011). *)
...
...
theories/decidable.v
View file @
c490d858
...
...
@@ 3,7 +3,7 @@
(** This file collects theorems, definitions, tactics, related to propositions
with a decidable equality. Such propositions are collected by the [Decision]
type class. *)
Require
Export
base
tactics
.
Require
Export
proof_irrel
.
Hint
Extern
200
(
Decision
_
)
=>
progress
(
lazy
beta
)
:
typeclass_instances
.
...
...
@@ 24,28 +24,38 @@ 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
.
Lemma
decide_true
{
A
}
`
{
Decision
P
}
(
x
y
:
A
)
:
P
→
(
if
decide
P
then
x
else
y
)
=
x
.
Proof
.
by
destruct
(
decide
P
).
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
.
(** The tactic [destruct_decide] destructs a sumbool [dec]. If one of the
components is double negated, it will try to remove the double negation. *)
Ltac
destruct_decide
dec
:
=
let
H
:
=
fresh
in
Tactic
Notation
"destruct_decide"
constr
(
dec
)
"as"
ident
(
H
)
:
=
destruct
dec
as
[
H

H
]
;
try
match
type
of
H
with

¬¬_
=>
apply
dec_stable
in
H
end
.
Tactic
Notation
"destruct_decide"
constr
(
dec
)
:
=
let
H
:
=
fresh
in
destruct_decide
dec
as
H
.
(** The tactic [case_decide] performs case analysis on an arbitrary occurrence
of [decide] or [decide_rel] in the conclusion or hypotheses. *)
Ltac
case_decide
:
=
Tactic
Notation
"case_decide"
"as"
ident
(
Hd
)
:
=
match
goal
with

H
:
context
[@
decide
?P
?dec
]

_
=>
destruct_decide
(@
decide
P
dec
)
destruct_decide
(@
decide
P
dec
)
as
Hd

H
:
context
[@
decide_rel
_
_
?R
?x
?y
?dec
]

_
=>
destruct_decide
(@
decide_rel
_
_
R
x
y
dec
)
destruct_decide
(@
decide_rel
_
_
R
x
y
dec
)
as
Hd


context
[@
decide
?P
?dec
]
=>
destruct_decide
(@
decide
P
dec
)
destruct_decide
(@
decide
P
dec
)
as
Hd


context
[@
decide_rel
_
_
?R
?x
?y
?dec
]
=>
destruct_decide
(@
decide_rel
_
_
R
x
y
dec
)
destruct_decide
(@
decide_rel
_
_
R
x
y
dec
)
as
Hd
end
.
Tactic
Notation
"case_decide"
:
=
let
H
:
=
fresh
in
case_decide
as
H
.
(** The tactic [solve_decision] uses Coq's [decide equality] tactic together
with instance resolution to automatically generate decision procedures. *)
...
...
@@ 107,23 +117,11 @@ Definition dexist `{∀ x : A, Decision (P x)} (x : A) (p : P x) : dsig P :=
x
↾
bool_decide_pack
_
p
.
Lemma
dsig_eq
`
(
P
:
A
→
Prop
)
`
{
∀
x
,
Decision
(
P
x
)}
(
x
y
:
dsig
P
)
:
x
=
y
↔
`
x
=
`
y
.
Proof
.
split
.
*
destruct
x
,
y
.
apply
proj1_sig_inj
.
*
intro
.
destruct
x
as
[
x
Hx
],
y
as
[
y
Hy
].
simpl
in
*.
subst
.
f_equal
.
revert
Hx
Hy
.
case
(
bool_decide
(
P
y
)).
+
by
intros
[]
[].
+
done
.
Qed
.
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
.
Global
Instance
dsig_eq_dec
`
(
P
:
A
→
Prop
)
`
{
∀
x
,
Decision
(
P
x
)}
`
{
∀
x
y
:
A
,
Decision
(
x
=
y
)}
(
x
y
:
dsig
P
)
:
Decision
(
x
=
y
).
Proof
.
refine
(
cast_if
(
decide
(
`
x
=
`
y
)))
;
by
rewrite
dsig_eq
.
Defined
.
(** * Instances of Decision *)
(** Instances of [Decision] for operators of propositional logic. *)
Instance
True_dec
:
Decision
True
:
=
left
I
.
...
...
@@ 164,3 +162,7 @@ Instance curry_dec `(P_dec : ∀ (x : A) (y : B), Decision (P x y)) p :
end
.
Instance
uncurry_dec
`
(
P_dec
:
∀
(
p
:
A
*
B
),
Decision
(
P
p
))
x
y
:
Decision
(
uncurry
P
x
y
)
:
=
P_dec
(
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
.
theories/list.v
View file @
c490d858
...
...
@@ 812,6 +812,9 @@ Proof.
revert
i
.
induction
n
;
intros
[?]
;
naive_solver
auto
with
lia
.
Qed
.
Lemma
replicate_S
n
(
x
:
A
)
:
replicate
(
S
n
)
x
=
x
::
replicate
n
x
.
Proof
.
done
.
Qed
.
Lemma
replicate_plus
n
m
(
x
:
A
)
:
replicate
(
n
+
m
)
x
=
replicate
n
x
++
replicate
m
x
.
Proof
.
induction
n
;
simpl
;
f_equal
;
auto
.
Qed
.
...
...
@@ 829,6 +832,14 @@ Lemma drop_replicate_plus n m (x : A) :
drop
n
(
replicate
(
n
+
m
)
x
)
=
replicate
m
x
.
Proof
.
rewrite
drop_replicate
.
f_equal
.
lia
.
Qed
.
Lemma
reverse_replicate
n
(
x
:
A
)
:
reverse
(
replicate
n
x
)
=
replicate
n
x
.
Proof
.
induction
n
as
[
n
IH
]
;
[
done
].
simpl
.
rewrite
reverse_cons
,
IH
.
change
[
x
]
with
(
replicate
1
x
).
by
rewrite
<
replicate_plus
,
plus_comm
.
Qed
.
(** ** Properties of the [resize] function *)
Lemma
resize_spec
(
l
:
list
A
)
n
x
:
resize
n
x
l
=
take
n
l
++
replicate
(
n

length
l
)
x
.
...
...
theories/numbers.v
View file @
c490d858
...
...
@@ 165,12 +165,26 @@ Notation "(<)" := Z.lt (only parsing) : Z_scope.
Infix
"`div`"
:
=
Z
.
div
(
at
level
35
)
:
Z_scope
.
Infix
"`mod`"
:
=
Z
.
modulo
(
at
level
35
)
:
Z_scope
.
Infix
"`quot`"
:
=
Z
.
quot
(
at
level
35
)
:
Z_scope
.
Infix
"`rem`"
:
=
Z
.
rem
(
at
level
35
)
:
Z_scope
.
Instance
Z_eq_dec
:
∀
x
y
:
Z
,
Decision
(
x
=
y
)
:
=
Z
.
eq_dec
.
Instance
Z_le_dec
:
∀
x
y
:
Z
,
Decision
(
x
≤
y
)%
Z
:
=
Z_le_dec
.
Instance
Z_lt_dec
:
∀
x
y
:
Z
,
Decision
(
x
<
y
)%
Z
:
=
Z_lt_dec
.
Instance
Z_inhabited
:
Inhabited
Z
:
=
populate
1
%
Z
.
(* Note that we cannot disable simpl for [Z.of_nat] as that would break
[omega] and [lia]. *)
Arguments
Z
.
to_nat
_
:
simpl
never
.
Arguments
Z
.
mul
_
_
:
simpl
never
.
Arguments
Z
.
add
_
_
:
simpl
never
.
Arguments
Z
.
opp
_
:
simpl
never
.
Arguments
Z
.
pow
_
_
:
simpl
never
.
Arguments
Z
.
div
_
_
:
simpl
never
.
Arguments
Z
.
modulo
_
_
:
simpl
never
.
Arguments
Z
.
quot
_
_
:
simpl
never
.
Arguments
Z
.
rem
_
_
:
simpl
never
.
(** * Notations and properties of [Qc] *)
Notation
"2"
:
=
(
1
+
1
)%
Qc
:
Qc_scope
.
Infix
"≤"
:
=
Qcle
:
Qc_scope
.
...
...
theories/proof_irrel.v
0 → 100644
View file @
c490d858
(* Copyright (c) 20122013, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects facts on proof irrelevant types/propositions. *)
Require
Export
Eqdep_dec
tactics
.
Hint
Extern
200
(
ProofIrrel
_
)
=>
progress
(
lazy
beta
)
:
typeclass_instances
.
Instance
:
ProofIrrel
True
.
Proof
.
by
intros
[]
[].
Qed
.
Instance
:
ProofIrrel
False
.
Proof
.
by
intros
[].
Qed
.
Instance
and_pi
(
A
B
:
Prop
)
:
ProofIrrel
A
→
ProofIrrel
B
→
ProofIrrel
(
A
∧
B
).
Proof
.
intros
??
[??]
[??].
by
f_equal
.
Qed
.
Instance
prod_pi
(
A
B
:
Type
)
:
ProofIrrel
A
→
ProofIrrel
B
→
ProofIrrel
(
A
*
B
).
Proof
.
intros
??
[??]
[??].
by
f_equal
.
Qed
.
Instance
eq_pi
{
A
}
`
{
∀
x
y
:
A
,
Decision
(
x
=
y
)}
(
x
y
:
A
)
:
ProofIrrel
(
x
=
y
).
Proof
.
intros
??.
apply
eq_proofs_unicity
.
intros
x'
y'
.
destruct
(
decide
(
x'
=
y'
))
;
tauto
.
Qed
.
Instance
Is_true_pi
(
b
:
bool
)
:
ProofIrrel
(
Is_true
b
).
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
.
*
destruct
x
,
y
.
apply
proj1_sig_inj
.
*
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
.
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