thery / poltac

Tactic for polynomial manipulations

Poltac

A set of tactics to deal with inequalities in Coq over N, Z and R:

• pols: simplification
• polf: factorization
• polr: replacement

inspired from https://hal.inria.fr/inria-00070394

To build:

make all

To use it:

Require Import PolTac.

Examples

See Nex.v, Zex.v and Rex.v.

Demo: a running example (polr_test2 in Rex.v)

1 subgoal
x : R
y : R
z : R
t : R
u : R
H : t < 0
H1 : y = u
H2 : x + z < y
______________________________________(1/1)
2 * y * t < x * t + t * u + z * t

We use polf to remove t from the left and the right side of the inequality since t is negative it changes the direction of the inequality.

polf.

1 subgoal

....
H1 : y = u
....
______________________________________(1/1)
x + u + z < 2 * y

We use polr to replace u by y in the goal.

polr H1; auto with real.

1 subgoal

....
H2 : x + z < y
....
______________________________________(1/1)
x + y + z < 2 * y

We use polr to bound x + y + z using H2.

polr H2.

2 subgoals
...
______________________________________(1/2)
x + y + z < y + y

...
______________________________________(2/2)
y + y <= 2 * y

We use pols to simply by y on both sides.

pols.

2 subgoals
...
______________________________________(1/2)
x + z < y

...
______________________________________(2/2)
y + y <= 2 * y

This exactly `H2`.

```coq
exact H2.

1 subgoal

...
______________________________________(1/1)
y + y <= 2 * y

We use pols to simply by y on both sides.

1 subgoal
...
______________________________________(1/1)
0 <= 0

---

[email protected]

Meta

• Author(s):
  • Laurent Théry
• License: MIT License
• Compatible Coq versions: 8.19 or later
• Additional dependencies: none
• Coq namespace: PolTac
• Related publication(s): none

Building and installation instructions

To build and install manually, do:

git clone https://github.com/thery/poltac.git
cd poltac
make   # or make -j <number-of-cores-on-your-machine> 
make install

poltac's Issues

Setting Z.add, Z.mul, etc as Opaque break the tactics

This is not very surprising, but setting Z.add or other operators to Opaque breaks PolTac (because it breaks tactics in poltac such as Zsimpl := simpl Z Zplus Zmult Zopp ...).

I'm not sure what is the proper way of fixing this.
If there was a way of ensuring that these identifiers are transparent locally, that would be a solution, but I don't think it's possible currently.
An other solution I see is rewriting Zsimpl and other to use rewriting with the right lemmas instead of simpl.