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DetermineValuesCase8.txt
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DetermineValuesCase8.txt
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====================================================================================================
CASE CONSIDERED: x~-10, 1+x~-1
====================================================================================================
GOAL: determine values of 1+kx for various integers k.
We'll do them one at a time.
====================================================================================================
*** Attempting to pin down value of 1+2x (mod squares). ***
This is what we know so far:
x ~ -10, x + 1 ~ -1.
Applying first-order constraints shows that the possible values of 2*x + 1 are: [5, -1].
Thus first-order constraints did not suffice to pin down the value uniquely.
Let's attempt second-order constraints as well.
----------------------------------------------------------------------------------------------------
Assume that 2*x + 1 ~ 5.
Given 2*x + 1 ~ 5, first-order constraints show that 3*x + 1 must be in [-5].
Let's consider each possibility in turn.
Assume 3*x + 1 ~ -5.
---> No contradictions obtained via second order constraints.
Hence the possibility that 2*x + 1 ~ 5 cannot be ruled out.
----------------------------------------------------------------------------------------------------
Assume that 2*x + 1 ~ -1.
Given 2*x + 1 ~ -1, first-order constraints show that 3*x + 1 must be in [2].
Let's consider each possibility in turn.
Assume 3*x + 1 ~ 2.
---> Contradiction obtained! 3*x + 1 can't be 2. Contradiction was found by considering 4*x + 1.
(In this case, we have the following assumptions in place:
{2*x + 1: -1, 3*x + 1: 2, x: -10, x + 1: -1}
Use the basic constraints on 4*x + 1 coming from these.)
Hence we have ruled out the possibility that 2*x + 1 ~ -1!
----------------------------------------------------------------------------------------------------
Conclusion: x ~ -10, x + 1 ~ -1 ==> 2*x + 1 ~ 5
====================================================================================================
====================================================================================================
*** Attempting to pin down value of 1+3x (mod squares). ***
This is what we know so far:
2*x + 1 ~ 5, x ~ -10, x + 1 ~ -1.
Applying first-order constraints shows that the possible values of 3*x + 1 are: [-5].
Didn't need to do anything extra!
----------------------------------------------------------------------------------------------------
Conclusion: x ~ -10, x + 1 ~ -1 ==> 3*x + 1 ~ -5.
====================================================================================================
*** Attempting to pin down value of 1+4x (mod squares). ***
This is what we know so far:
2*x + 1 ~ 5, 3*x + 1 ~ -5, x ~ -10, x + 1 ~ -1.
Applying first-order constraints shows that the possible values of 4*x + 1 are: [1].
Didn't need to do anything extra!
----------------------------------------------------------------------------------------------------
Conclusion: x ~ -10, x + 1 ~ -1 ==> 4*x + 1 ~ 1.
====================================================================================================
*** Attempting to pin down value of 1+5x (mod squares). ***
This is what we know so far:
2*x + 1 ~ 5, 3*x + 1 ~ -5, x ~ -10, x + 1 ~ -1, 4*x + 1 ~ 1.
Applying first-order constraints shows that the possible values of 5*x + 1 are: [-1].
Didn't need to do anything extra!
----------------------------------------------------------------------------------------------------
Conclusion: x ~ -10, x + 1 ~ -1 ==> 5*x + 1 ~ -1.
====================================================================================================
*** Attempting to pin down value of 1-2x (mod squares). ***
This is what we know so far:
5*x + 1 ~ -1, 2*x + 1 ~ 5, x ~ -10, 3*x + 1 ~ -5, x + 1 ~ -1, 4*x + 1 ~ 1.
Applying first-order constraints shows that the possible values of -2*x + 1 are: [5].
Didn't need to do anything extra!
----------------------------------------------------------------------------------------------------
Conclusion: x ~ -10, x + 1 ~ -1 ==> -2*x + 1 ~ 5.
====================================================================================================
*** Attempting to pin down value of 1-3x (mod squares). ***
This is what we know so far:
5*x + 1 ~ -1, 2*x + 1 ~ 5, x ~ -10, -2*x + 1 ~ 5, 3*x + 1 ~ -5, x + 1 ~ -1, 4*x + 1 ~ 1.
Applying first-order constraints shows that the possible values of -3*x + 1 are: [-1].
Didn't need to do anything extra!
----------------------------------------------------------------------------------------------------
Conclusion: x ~ -10, x + 1 ~ -1 ==> -3*x + 1 ~ -1.
====================================================================================================
*** Attempting to pin down value of 1-4x (mod squares). ***
This is what we know so far:
5*x + 1 ~ -1, 2*x + 1 ~ 5, x ~ -10, -2*x + 1 ~ 5, -3*x + 1 ~ -1, 3*x + 1 ~ -5, x + 1 ~ -1, 4*x + 1 ~ 1.
Applying first-order constraints shows that the possible values of -4*x + 1 are: [1].
Didn't need to do anything extra!
----------------------------------------------------------------------------------------------------
Conclusion: x ~ -10, x + 1 ~ -1 ==> -4*x + 1 ~ 1.
====================================================================================================
*** Attempting to pin down value of 1-5x (mod squares). ***
This is what we know so far:
5*x + 1 ~ -1, 2*x + 1 ~ 5, x ~ -10, -2*x + 1 ~ 5, -4*x + 1 ~ 1, -3*x + 1 ~ -1, 3*x + 1 ~ -5, x + 1 ~ -1, 4*x + 1 ~ 1.
Applying first-order constraints shows that the possible values of -5*x + 1 are: [-5].
Didn't need to do anything extra!
----------------------------------------------------------------------------------------------------
Conclusion: x ~ -10, x + 1 ~ -1 ==> -5*x + 1 ~ -5.
====================================================================================================
*** Attempting to pin down value of 1-x (mod squares). ***
This is what we know so far:
5*x + 1 ~ -1, 2*x + 1 ~ 5, x ~ -10, -2*x + 1 ~ 5, -4*x + 1 ~ 1, -3*x + 1 ~ -1, 3*x + 1 ~ -5, x + 1 ~ -1, -5*x + 1 ~ -5, 4*x + 1 ~ 1.
Applying first-order constraints shows that the possible values of -x + 1 are: [-5].
Didn't need to do anything extra!
----------------------------------------------------------------------------------------------------
Conclusion: x ~ -10, x + 1 ~ -1 ==> -x + 1 ~ -5.
====================================================================================================
*** Attempting to pin down value of 5+x (mod squares). ***
This is what we know so far:
5*x + 1 ~ -1, 2*x + 1 ~ 5, x ~ -10, -x + 1 ~ -5, -2*x + 1 ~ 5, -4*x + 1 ~ 1, -3*x + 1 ~ -1, 3*x + 1 ~ -5, x + 1 ~ -1, -5*x + 1 ~ -5, 4*x + 1 ~ 1.
First-order constraints show that 5+x ~ -5
----------------------------------------------------------------------------------------------------
Conclusion: x ~ -10, x + 1 ~ -1 ==> x + 5 ~ -5
==> x/5 + 1 ~ -1.
====================================================================================================
*** Attempting to pin down value of -5+x (mod squares). ***
This is what we know so far:
5*x + 1 ~ -1, 2*x + 1 ~ 5, x/5 + 1 ~ -1, x ~ -10, -x + 1 ~ -5, -2*x + 1 ~ 5, -4*x + 1 ~ 1, -3*x + 1 ~ -1, 3*x + 1 ~ -5, x + 1 ~ -1, -5*x + 1 ~ -5, 4*x + 1 ~ 1.
First-order constraints show that -5+x ~ 1
----------------------------------------------------------------------------------------------------
Conclusion: x ~ -10, x + 1 ~ -1 ==> x - 5 ~ 1
==> -x/5 + 1 ~ -5.
====================================================================================================
====================================================================================================
In summary, we have that in this case, norm combinatorics implies:
5*x + 1 ~ -1
2*x + 1 ~ 5
x/5 + 1 ~ -1
-x + 1 ~ -5
-2*x + 1 ~ 5
-4*x + 1 ~ 1
-3*x + 1 ~ -1
3*x + 1 ~ -5
-x/5 + 1 ~ -5
-5*x + 1 ~ -5
4*x + 1 ~ 1
====================================================================================================