Is nested induction necessary for all variables

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Let $S(n_1,n_2,n_3)$ be the statement to prove in natural numbers;



If I prove the following three



  1. $S(1,1,1)$ is true


  2. $forall n_1n_2 S(n_1,n_2,n_3) implies S(n_1,n_2,n_3+1)$


  3. $forall n_1n_3 S(n_1,n_2,n_3) implies S(n_1,n_2+1,n_3)$


Now, Is it sufficient or we have to prove for $n_1$ also?




induction




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    up vote
    3
    down vote

    favorite
    1












    Let $S(n_1,n_2,n_3)$ be the statement to prove in natural numbers;



    If I prove the following three



    1. $S(1,1,1)$ is true


    2. $forall n_1n_2 S(n_1,n_2,n_3) implies S(n_1,n_2,n_3+1)$


    3. $forall n_1n_3 S(n_1,n_2,n_3) implies S(n_1,n_2+1,n_3)$


    Now, Is it sufficient or we have to prove for $n_1$ also?




    induction




    share|cite|improve this question























      up vote
      3
      down vote

      favorite
      1









      up vote
      3
      down vote

      favorite
      1






      1





      Let $S(n_1,n_2,n_3)$ be the statement to prove in natural numbers;



      If I prove the following three



      1. $S(1,1,1)$ is true


      2. $forall n_1n_2 S(n_1,n_2,n_3) implies S(n_1,n_2,n_3+1)$


      3. $forall n_1n_3 S(n_1,n_2,n_3) implies S(n_1,n_2+1,n_3)$


      Now, Is it sufficient or we have to prove for $n_1$ also?




      induction




      share|cite|improve this question













      Let $S(n_1,n_2,n_3)$ be the statement to prove in natural numbers;



      If I prove the following three



      1. $S(1,1,1)$ is true


      2. $forall n_1n_2 S(n_1,n_2,n_3) implies S(n_1,n_2,n_3+1)$


      3. $forall n_1n_3 S(n_1,n_2,n_3) implies S(n_1,n_2+1,n_3)$


      Now, Is it sufficient or we have to prove for $n_1$ also?





      induction





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      edited Aug 2 at 23:12
























      asked Aug 2 at 23:07









      hanugm

      789419




      789419




















          2 Answers
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          accepted










          Yes, you need one for $n_1$ as well. As it stands, you don't even have any way of knowing whether $S(2,1,1)$ is true from those assumptions.






          share|cite|improve this answer





















          • For example, if $S(n_1, n_2, n_3)$ is the proposition $n_1 = 1$ then the conditions hold...
            – Daniel Schepler
            Aug 2 at 23:16

















          up vote
          1
          down vote













          You have to prove it for $n_1$ as well. Consider the statement:
          $$
          S(n_1,n_2,n_3) leftrightarrow
          (n_1 < 2) land (n_2 = n_2) land (n_3 = n_3)
          $$
          We have that $S(1,1,1)$ is true. For $n_1 > 1$, the statement is false, so the implications are all true; for $n_1=1$, the statement is true regardless of $n_2$ and $n_3$, so the implications are still true.



          Note: I'm not sure if this is sufficient to prove $S(n_1,n_2,n_3)$ for every $n_1,n_2,n_3$. We'd have to prove it formally.






          share|cite|improve this answer





















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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            3
            down vote



            accepted










            Yes, you need one for $n_1$ as well. As it stands, you don't even have any way of knowing whether $S(2,1,1)$ is true from those assumptions.






            share|cite|improve this answer





















            • For example, if $S(n_1, n_2, n_3)$ is the proposition $n_1 = 1$ then the conditions hold...
              – Daniel Schepler
              Aug 2 at 23:16














            up vote
            3
            down vote



            accepted










            Yes, you need one for $n_1$ as well. As it stands, you don't even have any way of knowing whether $S(2,1,1)$ is true from those assumptions.






            share|cite|improve this answer





















            • For example, if $S(n_1, n_2, n_3)$ is the proposition $n_1 = 1$ then the conditions hold...
              – Daniel Schepler
              Aug 2 at 23:16












            up vote
            3
            down vote



            accepted







            up vote
            3
            down vote



            accepted






            Yes, you need one for $n_1$ as well. As it stands, you don't even have any way of knowing whether $S(2,1,1)$ is true from those assumptions.






            share|cite|improve this answer













            Yes, you need one for $n_1$ as well. As it stands, you don't even have any way of knowing whether $S(2,1,1)$ is true from those assumptions.







            share|cite|improve this answer













            share|cite|improve this answer



            share|cite|improve this answer











            answered Aug 2 at 23:10









            Arthur

            98.1k792173




            98.1k792173











            • For example, if $S(n_1, n_2, n_3)$ is the proposition $n_1 = 1$ then the conditions hold...
              – Daniel Schepler
              Aug 2 at 23:16
















            • For example, if $S(n_1, n_2, n_3)$ is the proposition $n_1 = 1$ then the conditions hold...
              – Daniel Schepler
              Aug 2 at 23:16















            For example, if $S(n_1, n_2, n_3)$ is the proposition $n_1 = 1$ then the conditions hold...
            – Daniel Schepler
            Aug 2 at 23:16




            For example, if $S(n_1, n_2, n_3)$ is the proposition $n_1 = 1$ then the conditions hold...
            – Daniel Schepler
            Aug 2 at 23:16










            up vote
            1
            down vote













            You have to prove it for $n_1$ as well. Consider the statement:
            $$
            S(n_1,n_2,n_3) leftrightarrow
            (n_1 < 2) land (n_2 = n_2) land (n_3 = n_3)
            $$
            We have that $S(1,1,1)$ is true. For $n_1 > 1$, the statement is false, so the implications are all true; for $n_1=1$, the statement is true regardless of $n_2$ and $n_3$, so the implications are still true.



            Note: I'm not sure if this is sufficient to prove $S(n_1,n_2,n_3)$ for every $n_1,n_2,n_3$. We'd have to prove it formally.






            share|cite|improve this answer

























              up vote
              1
              down vote













              You have to prove it for $n_1$ as well. Consider the statement:
              $$
              S(n_1,n_2,n_3) leftrightarrow
              (n_1 < 2) land (n_2 = n_2) land (n_3 = n_3)
              $$
              We have that $S(1,1,1)$ is true. For $n_1 > 1$, the statement is false, so the implications are all true; for $n_1=1$, the statement is true regardless of $n_2$ and $n_3$, so the implications are still true.



              Note: I'm not sure if this is sufficient to prove $S(n_1,n_2,n_3)$ for every $n_1,n_2,n_3$. We'd have to prove it formally.






              share|cite|improve this answer























                up vote
                1
                down vote










                up vote
                1
                down vote









                You have to prove it for $n_1$ as well. Consider the statement:
                $$
                S(n_1,n_2,n_3) leftrightarrow
                (n_1 < 2) land (n_2 = n_2) land (n_3 = n_3)
                $$
                We have that $S(1,1,1)$ is true. For $n_1 > 1$, the statement is false, so the implications are all true; for $n_1=1$, the statement is true regardless of $n_2$ and $n_3$, so the implications are still true.



                Note: I'm not sure if this is sufficient to prove $S(n_1,n_2,n_3)$ for every $n_1,n_2,n_3$. We'd have to prove it formally.






                share|cite|improve this answer













                You have to prove it for $n_1$ as well. Consider the statement:
                $$
                S(n_1,n_2,n_3) leftrightarrow
                (n_1 < 2) land (n_2 = n_2) land (n_3 = n_3)
                $$
                We have that $S(1,1,1)$ is true. For $n_1 > 1$, the statement is false, so the implications are all true; for $n_1=1$, the statement is true regardless of $n_2$ and $n_3$, so the implications are still true.



                Note: I'm not sure if this is sufficient to prove $S(n_1,n_2,n_3)$ for every $n_1,n_2,n_3$. We'd have to prove it formally.







                share|cite|improve this answer













                share|cite|improve this answer



                share|cite|improve this answer











                answered Aug 2 at 23:17









                Sambo

                1,2771427




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