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You may be proving a lemma, a theorem, a corollary, etc
A proof is based on:
definition(s)
axioms
postulates
rules of inference (typical normal logic and mathematics)
To be accepted as "true" or "valid"
Recognized people in the field need to agree your
definitions are reasonable
axioms, postulates, ... are reasonable
rules of inference are reasonable and correctly applied
"True" and "Valid" are human intuitive judgments but can be
based on solid reasoning as presented in a proof.
Types of proofs include:
Direct proof (typical in Euclidean plane geometry proofs)
Write down line by line provable statements,
(e.g. definition, axiom, statement that follows from applying
the axiom to the definition, statement that follows from
applying a rule of inference from prior lines, etc.)
Proof by contradiction:
Given definitions, axioms, rules of inference
Assume Statement_A
use proof technique to derive a contradiction
(e.g. prove not Statement_A or prove Statement_B = not Statement_B,
like 1 = 2 or n > 2n)
Proof by induction (on Natural numbers)
Given a statement based on, say n, where n ranges over natural numbers
Prove the statement for n=0 or n=1
a) Prove the statement for n+1 assuming the statement true for n
b) Prove the statement for n+1 assuming the statement true for n in 1..n
Prove two sets A and B are equal, prove part 1, A is a subset of B
prove part 2, B is a subset of A
Prove two machines M1 and M2 are equal,
prove part 1 that machine M1 can simulate machine M2
prove part 2 that machine M2 can simulate machine M1
Limits on proofs:
Godel incompleteness theorem:
a) Any formal system with enough power to handle arithmetic will
have true theorems that are unprovable in the formal system.
(He proved it with Turing machines.)
b) Adding axioms to the system in order to be able to prove all the
"true" (valid) theorems will make the system "inconsistent."
Inconsistent means a theorem can be proved that is not accepted
as "true" (valid).
c) Technically, any formal system with enough power to do arithmetic
is either incomplete or inconsistent.
For reference, read through the automata definitions and
language definitions.
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