.. raw:: latex

    \mainmatter

.. _proofs_by_calculation:
    
Proofs by calculation
=====================

This book begins in the familiar world of numbers: :math:`\mathbb{N}`, the natural
numbers (which in this book include 0);
:math:`\mathbb{Z}`, the integers; :math:`\mathbb{Q}`, the rational numbers;
and :math:`\mathbb{R}`, the real numbers.  We solve problems which feel pretty close
to high school algebra -- deducing equalities/inequalities from other
equalities/inequalities -- using a technique which is not usually taught in high
school algebra: building a single chain of expressions connecting the left-hand
side with the right.

.. include:: ch01_Proofs_by_Calculation/01_Proving_Equalities.inc
.. include:: ch01_Proofs_by_Calculation/02_Proving_Equalities_in_Lean.inc
.. include:: ch01_Proofs_by_Calculation/03_Tips_and_Tricks.inc
.. include:: ch01_Proofs_by_Calculation/04_Proving_Inequalities.inc
.. include:: ch01_Proofs_by_Calculation/05_A_Shortcut.inc