characterized by rigor; rigidly severe or harsh, as people, rules, or discipline: rigorous laws.
severely exact or accurate; precise: rigorous research.
(of weather or climate) uncomfortably severe or harsh; extremely inclement.
Logic, Mathematics. logically valid.

1350–1400; Middle English < Medieval Latin rigōrōsus. See rigor, -ous

rigorously, adverb
rigorousness, noun
overrigorous, adjective
overrigorously, adverb
overrigorousness, noun
self-rigorous, adjective
semirigorous, adjective
semirigorously, adverb
semirigorousness, noun
unrigorous, adjective
unrigorously, adverb
unrigorousness, noun

1. stern, austere, hard, inflexible, stiff, unyielding. See strict. 2. demanding, finical. 3. hard, bitter.

1. flexible, soft. 2. inaccurate. 3. mild.
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Based on the Random House Dictionary, © Random House, Inc. 2014.
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World English Dictionary
rigorous (ˈrɪɡərəs)
1.  characterized by or proceeding from rigour; harsh, strict, or severe: rigorous discipline
2.  severely accurate; scrupulous: rigorous book-keeping
3.  (esp of weather) extreme or harsh
4.  maths, logic (of a proof) making the validity of the successive steps completely explicit

Collins English Dictionary - Complete & Unabridged 10th Edition
2009 © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins
Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009
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Word Origin & History

late 14c., from O.Fr. rigorous (Mod.Fr. rigoureux), from M.L. rigorosus, from L. rigor (see rigor).
Online Etymology Dictionary, © 2010 Douglas Harper
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Example sentences
Decisions made at the top are carried out rigorously to the lowest level.
The elements of the word are related to each other in a specific way and follow
  each other in a rigorously determined sequence.
Every day his arguments became more extreme, more rigorously exact, and more
  distressing to his master.
We certainly need teachers who are both highly skilled and who can think
  rigorously beyond the confines of their own discipline.
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