## Overview

This library provides vernacular files containing tactics for simplifying the proofs of inequalities on expressions of real numbers for the Coq proof assistant.

This package is free software; you can redistribute it and/or
modify it under the terms of CeCILL-C Free Software License (see
the `COPYING`

file in the archive).

This library is hosted on the Inria Gitlab server. It is mainly developed by Guillaume Melquiond. Automatically generated documentation of the vernacular files is available here.

## Building and Installing using OPAM

If you are managing your Coq installation using Opam, you can install the library using the following command:

opam install --jobs=2 coq-interval

Note that the `coq-interval`

package is hosted in the Opam
repository dedicated
to Coq libraries.
So you have to type the following command beforehand, if your Opam
installation does not yet know about this repository.

opam repo add coq-released https://coq.inria.fr/opam/released

## Building and Installing from Sources

### Source tarballs

The following is a selection of some releases of the library, depending on the version of Coq. Minimal versions required for the dependencies are also indicated.

- version 4.11.0 (Coq 8.13 to 8.19, Flocq 3.2, MathComp 1.9, Coquelicot 3.1),
- version 4.10.0 (Coq 8.11 to 8.19, Flocq 3.2, MathComp 1.9, Coquelicot 3.1),
- version 4.6.1 (Coq 8.8 to 8.17, Flocq 3.1, MathComp 1.7, Coquelicot 3.0),
- version 3.4.2 (Coq 8.7 to 8.11, Flocq 3.0, MathComp 1.6, Coquelicot 3.0),
- version 3.3.0 (Coq 8.5 to 8.8, Flocq 2.5, MathComp 1.6, Coquelicot 3.0),
- version 3.1.1 (Coq 8.5 & 8.6, Flocq 2.5, MathComp 1.6, Coquelicot),
- version 2.2.1 (Coq 8.4 & 8.5, Flocq 2.5, MathComp 1.6, Coquelicot).

Other releases can be found here.

### Dependencies

In addition to the Coq proof assistant, you need the following libraries:

- Mathematical Components,
- Flocq,
- Coquelicot,
- BigNums (for Coq >= 8.7).

### Configuring, compiling, and installing

Ideally, you should just have to type:

./configure && ./remake --jobs=2 && ./remake install

The environment variable `COQC`

can be passed to the
configure script in order to set the Coq compiler command. The configure
script defaults to `coqc`

. Similarly, `COQDEP`

can
be used to specify the location of `coqdep`

. The
`COQBIN`

environment variable can be used to set both
variables at once.

The library files are compiled at the logical location
`Interval`

. The `COQUSERCONTRIB`

environment
variable can be used to override the physical location where
the `Interval`

directory containing these files will be
installed by `./remake install`

. By default, the target
directory is ``$COQC -where`/user-contrib`

.

## Invocation

In order to use the tactics of the library, one has to import the
`Interval.Tactic`

file into a Coq proof script. The main
tactic is named `interval`

.

The tactic can be applied on a goal of the form ```
c1 <= e
<= c2
```

with `e`

an expression involving
real-valued operators. Sub-expressions that are not recognized by the
tactic should be either terms `t`

appearing in hypothesis
inequalities `c3 <= t <= c4`

or simple integers. The
bounds `c1`

, `c2`

, etc, are expressions that contain
only constant leaves, e.g., `5 / sqrt (1 + PI)`

.

The complete list of recognized goals is as follows:

`c1 <= e <= c2`

;`e <= c1`

,`c1 <= e`

,`e >= c1`

, and`c1 >= e`

;`e < c1`

,`c1 < e`

,`e > c1`

, and`c1 > e`

;`e <> c1`

and`c1 <> e`

;`Rabs e <= c1`

, handled as`-c1 <= e <= c1`

.

The complete list of recognized hypotheses is as follows:

`c1 <= t <= c2`

,`c1 <= t < c2`

,`c1 < t <= c2`

,`c1 < t < c2`

;`t <= c1`

,`c1 <= t`

,`t >= c1`

, and`c1 >= t`

;`t < c1`

,`c1 < t`

,`t > c1`

, and`c1 > t`

;`Rabs t <= c1`

and`Rabs t < c1`

, handled as`-c1 <= e <= c1`

.

The tactics recognize the following operators: `PI`

, `Ropp`

,
`Rabs`

, `Rinv`

, `Rsqr`

, `sqrt`

,
`cos`

, `sin`

, `tan`

, `atan`

,
`exp`

, `ln`

, `pow`

, `Rpower`

, `powerRZ`

,
`Rplus`

, `Rminus`

, `Rmult`

, `Rdiv`

.
Flocq's operators `Zfloor`

, `Zceil`

, `Ztrunc`

,
`ZnearestE`

(composed with `IZR`

) are also
recognized. Flocq's operator `Generic_fmt.round`

with format
`FLT_exp`

is also supported.
There are some restrictions on the domain of a few
functions: `pow`

and `powerRZ`

should be written
with a numeric exponent; the input of `cos`

and `sin`

should be between `-2*PI`

and `2*PI`

; the input of `tan`

should be between
`-PI/2`

and `PI/2`

. Outside of these domains, the
trigonometric functions return tight results only for singleton
input intervals.

A helper tactic `interval_intro e`

is also available. Instead
of proving the current goal, it computes an enclosure of the expression
`e`

passed as argument and it introduces the resulting
inequalities into the proof context. If only one bound is needed, the
keywords `lower`

and `upper`

can be passed to the
tactic, so that it does not perform useless computations. For
example, `interval_intro e lower`

introduces only the
inequality corresponding to the lower bound of `e`

in the
context. The `interval_intro`

tactic uses a fresh name for the
generated inequalities, unless one uses `as`

followed by an
intro pattern.

The `integral`

tactic is a specialized version
of `interval`

that can be used when the target enclosure
involves an expression containing an integral. Such an integral should be expressed
using `RInt`

; its bounds should be constant; and its integrand
should be an expression containing only constant leaves except for the
integration variable. Improper integrals are also supported, when
expressed using `RInt_gen`

. The supported bounds are
then `(at_right 0) (at_point _)`

and ```
(at_point _)
(Rbar_locally p_infty)
```

. In the improper case, the integrand should
be of the form `(fun t => f t * g t)`

with `f`

a function bounded on the integration domain and `g t`

one of
the following expressions:

`(ln t) ^ _`

,`powerRZ t _ * (ln t) ^ _`

,`/ (t * (ln t) ^ _)`

.

The `root H`

tactic is a specialized version
of `interval`

that can be used when the target enclosure
involves an expression that is constrained by an equation whose proof
is `H`

. Rather than an equality proof, the tactic can also
be passed an equality statement or even just a real expression (then
assumed to be zero), in which case the tactic generates a new goal
that asks for a proof of this equality.

The helper tactic `integral_intro`

is the counterpart
of `interval_intro`

, but for introducing enclosures of
integrals into the proof context. As with `interval_intro`

,
keywords `lower`

, `upper`

, and `as`

, are
supported. Similarly, `root_intro`

introduces enclosures of roots into
the proof context. Only `as`

is supported in that case. When an
equality (rather than a proof) is passed to `root_intro`

, no new goal
is generated, but the introduced hypothesis uses the equality as its
antecedent.

Tactics `interval_intro`

, `integral_intro`

,
and `root_intro`

are available as degenerate forms that
unify the obtained enclosure with the current goal. They are meant to
be used when the goal is an existential variable, e.g., in a
tactic-in-term context. Those degenerate forms are
named `interval`

, `integral`

,
and `root`

, for conciseness, but they should not be
confused with the original tactics. For all intents and purposes, they
behave like the `*_intro`

tactics.

Both `interval_intro`

and `integral_intro`

are available as degenerate
forms that unify the obtained enclosure with the current goal. They
are meant to be used when the goal is an existential variable, e.g.,
in a tactic-in-term context. Those degenerate forms are named
`interval`

and `integral`

for conciseness, but
they should not be confused with the original tactics. For all intents
and purposes, they behave like `interval_intro`

and `integral_intro`

.

The `plot`

tactic produces correct function graphs, that
is, two curves that are guaranteed to enclose the given function on the
given input interval. It is invoked as `plot f x1 x2`

. An
output range can optionally be passed: `plot f x1 x2 y1 y2`

.
If not, it is computed on the fly.

The `Interval.Plot`

file provides a `Plot`

command that can be used to display a function graph. This is done by
invoking the `gnuplot`

command, which thus needs to be
installed. If a string is passed as an optional argument, instead of
invoking `gnuplot`

, a file is created with the corresponding
Gnuplot script.

This file also provides a `Def`

command, which has a syntax
similar to the `Definition`

, except that the right-hand
side is not a Gallina term but an Ltac term. This command is meant to
be used in conjunction with the degenerate forms of the
tactics `interval`

, `integral`

,
and `root`

. If the name given to the term does not matter,
the simpler `Do`

command can be used instead. Both commands
display the type of the generated term with some syntactic sugar. If
the type is a function graph (as produced by `plot`

), then
the command `Plot`

is invoked on it.

## Fine-tuning

The behavior of the tactics can be tuned by passing an optional set of
parameters `with (param1, param2, ...)`

. These parameters are
parsed from left to right. If some parameters are conflicting, the
earlier ones are discarded. Available parameters are as follows (with the
type of their arguments, if any):

`i_prec (p:positive)`

- Set the precision used to emulate floating-point computations. If
this parameter is not specified, the tactics perform computations
using machine floating-point numbers, when available. Otherwise, the
tactic defaults to using
`i_prec 53`

. Note that, in some corner cases, the tactics might fail when using native numbers, despite the goals being provable using a 53-bit emulation. `i_native_compute`

- Perform computations using
`native_compute`

instead of`vm_compute`

. This greatly increases the startup time of the tactics, but makes the computations faster. This is useful only for computationally-intensive proofs. `i_bisect (x:R)`

- Instruct the tactics to split the interval enclosing
`x`

until the goal is proved on all the sub-intervals. Several`i_bisect`

parameters can be given. In that case, the tactic cycles through all of them, splitting the input domain along the corresponding variable. Computation time is more or less proportional to the final number of sub-domains. This parameter is only meaningful for the`interval`

and`interval_intro`

tactics. `i_depth (n:nat)`

- Set the maximal bisection depth. Setting it to a nonzero value has
no effect unless
`i_bisect`

parameters are also passed. If the maximal depth is`n`

, the tactic will consider up to 2^{n}sub-domains in the worst case. As with`i_bisect`

, this parameter is only meaningful for the`interval`

and`interval_intro`

tactics. The maximal depth defaults to`15`

for`interval`

, and to`5`

for`interval_intro`

. Note that`interval_intro`

computes the best enclosure that could be verified by`interval`

using the same maximal depth. `i_autodiff (x:R)`

- Instruct the tactics to perform an automatic differentiation of the
target expression with respect to
`x`

. This makes the tactic about twice slower on each sub-domain. But it makes it possible to detect some monotony properties of the target expression, thus reducing the amount of sub-domains that need to be considered. Note that this is only useful if there are several occurrences of`x`

in the goal. This parameter is only meaningful for the`interval`

and`interval_intro`

tactics. It is mutually exclusive with`i_taylor`

. This parameter can also be used with the`root`

and`root_intro`

tactics, if they did not guess the correct target variable. `i_taylor (x:R)`

- Instruct the tactics to compute a reliable polynomial enclosure of
the target expression using Taylor models in
`x`

. As with`i_autodiff`

, this is useful only if`x`

occurs several times in the goal. Computing polynomial enclosures is much slower than automatic differentiation, but it can reduce the final number of sub-domains even further, thus speeding up proofs. Note that it might fail to prove goals that are feasible using automatic differentiation. As with`i_autodiff`

, the`i_taylor`

parameter is only meaningful for the`interval`

and`interval_intro`

tactics. It is implicit for the`integral`

,`integral_intro`

, and`plot`

tactics, as Taylor models of the integrand (respectively, plotted function) are computed with respect to its variable. `i_degree (d:nat)`

- Set the degree of polynomials used as enclosures. The default degree
is 10. For
`interval`

and`interval_intro`

, this parameter is only meaningful in conjunction with`i_taylor`

. `i_fuel (n:positive)`

- Set the maximum number of sub-domains considered when bounding
integrals. The tactics maintain a set of integration sub-domains; it
splits the sub-domains that contribute the most to the inaccuracy of
the integral until its enclosure is tight enough to satisfy the goal.
By default, the tactics will split the integration domain into at
most 100 sub-domains. This parameter is only meaningful for
the
`integral`

and`integral_intro`

tactics. `i_width (p:Z)`

- Instruct the
`integral_intro`

tactic to compute an enclosure of the integral that is no larger than 2^{p}. The tactic will split the integration domain until the resulting enclosure reaches this width or`i_fuel`

is exhausted. This parameter is meaningless for the other tactics. It is mutually exclusive with`i_relwidth`

. `i_relwidth (p:positive)`

- Instruct the
`integral_intro`

tactic to compute an enclosure of the integral whose relative width is no larger than 2^{-p}. This parameter is meaningless for the other tactics. It defaults to 10. This means that, if neither`i_width`

nor`i_relwidth`

is used,`integral_intro`

will compute an enclosure of the integral accurate to three decimal digits, assuming`i_fuel`

is large enough. `i_size (w h:positive)`

- Instruct the
`plot`

tactic to target a resolution of`w`

by`h`

pixels. This parameter is meaningless for the other tactics. It defaults to a resolution of`512x384`

. The tactic will subdivide the input interval into`w`

subintervals, and it will try to ensure that the function graph is no larger than a few pixels vertically. `i_delay`

- Prevent Coq from verifying the generated proof at invocation time.
Instead, Coq will check the proof term at
`Qed`

time. This makes the tactics`interval`

,`integral`

, and`root`

instant. But it also means that failures, if any, will only be detected at`Qed`

time, possibly with an inscrutable error message. This parameter is thus meant to be used when editing a proof script for which the tactics are already known to succeed. For the tactics`interval_intro`

,`integral_intro`

, and`root_intro`

, computations are performed anyway (the risk of failure is thus negligible), but the`i_delay`

parameter postpones their verification until`Qed`

time. This makes these tactics twice as fast and is especially useful when optimizing the arguments of`i_prec`

,`i_degree`

, etc. For the degenerate forms of`interval_intro`

,`integral_intro`

, and`root_intro`

, the`i_delay`

parameter is always passed implicitly. `i_decimal`

- Instruct the tactics to output interval bounds using a decimal
representation. This parameter is only meaningful for the tactics
`interval_intro`

,`integral_intro`

, and`root_intro`

, as well as the corresponding degenerate tactics.

## Examples

From Coq Require Import Reals Lra. From Interval Require Import Tactic. Open Scope R_scope. Notation "x = y ± z" := (Rle (Rabs (x - y)) z) (at level 70, y at next level). (* Tactic interval *) Goal forall x, -1 <= x <= 1 -> sqrt (1 - x) <= 3/2. Proof. intros. interval. Qed. Goal forall x, -1 <= x <= 1 -> sqrt (1 - x) <= 141422/100000. Proof. intros. interval. Qed. Goal forall x, -1 <= x <= 1 -> sqrt (1 - x) <= 141422/100000. Proof. intros. interval_intro (sqrt (1 - x)) upper as H'. apply Rle_trans with (1 := H'). lra. Qed. Goal forall x, 3/2 <= x <= 2 -> forall y, 1 <= y <= 33/32 -> sqrt(1 + x/sqrt(x+y)) = 144/1000*x + 118/100 ± 71/32768. Proof. intros. interval with (i_prec 19, i_bisect x). Qed. Goal forall x, 1/2 <= x <= 2 -> sqrt x = ((((122 / 7397 * x + (-1733) / 13547) * x + 529 / 1274) * x + (-767) / 999) * x + 407 / 334) * x + 227 / 925 ± 5/65536. Proof. intros. interval with (i_bisect x, i_taylor x, i_degree 3). Qed. Goal forall x, -1 <= x -> x < 1 + powerRZ x 3. Proof. intros. apply Rminus_lt. interval with (i_bisect x, i_autodiff x). Qed. From Flocq Require Import Core. Notation round := (round radix2 (FLT_exp (-1074) 53) ZnearestE). Goal forall x, -1 <= x <= 1 -> rnd (1 + rnd (x * rnd (1 + rnd (x * (922446257493983/2251799813685248))))) = exp x ± 31/100. Proof. intros. interval with (i_taylor x). Qed. (* Tactic integral *) From Coquelicot Require Import Coquelicot. Goal RInt (fun x => atan (sqrt (x*x + 2)) / (sqrt (x*x + 2) * (x*x + 1))) 0 1 = 5/96*PI*PI ± 1/1000. Proof. integral with (i_fuel 2, i_degree 5). Qed. Goal RInt_gen (fun x => 1 * (powerRZ x 3 * ln x^2)) (at_right 0) (at_point 1) = 1/32. Proof. refine ((fun H => Rle_antisym _ _ (proj2 H) (proj1 H)) _). integral with (i_prec 10). Qed. (* Tactic root *) Goal forall x:R, 999 <= x <= 1000 -> sin x = 0 -> x = 318 * PI ± 1/1000. Proof. intros x Hx Hs. root Hs. Qed. (* Degenerate forms *) Definition equal_1 x `(0 <= x <= PI/2) := ltac:(interval ((cos x)² + (sin x)²) with (i_taylor x)). Definition equal_PI_over_4 := ltac:(integral (RInt (fun x => 1 / (1+x*x)) 0 1)). Definition equal_0_442854401002 x := ltac:(root (exp x = 2 - x) with i_decimal). (* Tactic plot and command Plot *) From Interval Require Import Plot. Definition p1 := ltac:(plot (fun x => x^2 * sin (x^2)) (-4) 4). Plot p1. Definition p2 := ltac:( plot (fun x => sin (x + exp x)) 0 6 (-5/4) (5/4) with (i_size 120 90, i_degree 6)). Plot p2 as "picture.gnuplot". Plot ltac:(plot (fun x => sqrt (1 - x^2) * sin (x * 200)) (-1) 1 with (i_degree 1, i_size 100 300)). (* Commands Do and Def *) Do interval (PI²/6). Do integral (RInt_gen (fun x => 1/(1 + x)^2 * (ln x)^2) (at_right 0) (at_point 1)) with (i_relwidth 30). Def quintic x := root (x^5 - x = 1).