# Prepare this notebook for defining the theorems of a theory:
% theorems_notebook # Keep this at the top following 'import proveit'.
from proveit.logic import Not , FALSE , Forall , Implies , Equals , TRUE , Or , NotEquals , Boolean , in_bool
from proveit import A , B
% begin theorems
Defining theorems for theory 'proveit.logic.booleans.negation'
Subsequent end-of-cell assignments will define theorems
'%end theorems' will finalize the definitions
negation_intro = Forall ( A , Not ( A ), conditions = [ Equals ( A , FALSE )])
negation_contradiction = Forall ( A , FALSE , conditions = [ A , Not ( A )])
untrue_from_negation = Forall ( A , NotEquals ( A , TRUE ), conditions = [ Not ( A )])
falsified_negation_intro = Forall ( A , Equals ( Not ( A ), FALSE ), conditions = [ A ])
double_negation_intro = Forall ( A , Not ( Not ( A )), conditions = [ A ])
double_negation_elim_lemma = Forall ( A , A , conditions = [ Not ( Not ( A ))],
domain = Boolean )
double_negation_elim = Forall ( A , A , conditions = [ Not ( Not ( A ))])
closure = Forall ( A , in_bool ( Not ( A )), domain = Boolean )
closure (conjecture with conjecture-based proof):
double_neg_closure = Forall ( A , in_bool ( Not ( Not ( A ))), domain = Boolean )
double_negation_equiv = Forall ( A , Equals ( Not ( Not ( A )), A ), domain = Boolean )
These theorems may now be imported from the theory package: proveit.logic.booleans.negation
These web pages were generated on 2021-01-19 by
Prove-It Beta Version 0.3, licensed under the GNU Public License by Sandia Corporation.
Presented proofs are not absolutely guaranteed. For assurance, it is important to check the structure
of the statement being proven, independently verify the derivation steps, track dependencies, and ensure that
employed axioms are valid and properly structured. Inconsistencies may exist, unknowingly, in this system.
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under the Quantum Computing Application Teams program. Sandia National Labs is managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a subsidiary of Honeywell International, Inc., for the U.S. Dept. of Energy's NNSA under contract DE-NA0003525. The views expressed above do not necessarily represent the views of the DOE or the U.S. Government.
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