Re: [isabelle] Proving two real^3 spaces are the same



On Wed, 30 Mar 2011 07:33:25 s.wong.731 at gmail.com wrote:
> Hi,
> 
> If I have two real^3 spaces S1 and S2 as:
> 
> axiomatization
> S1 :: "real ^ 2" and
> S2 :: "real ^ 2" where
> r1: "S1 $ 0 = S2 $ 0" and
> r2: "S1 $ 1 = S2 $ 1" and
> r3: "S1 $ 2 = S2 $ 2"

You've written "real ^ 2" where I think you mean "real ^ 3", but 
I'll leave it as it is for illustrative purposes.

> lemma "S1 = S2"
> 
> How would one go about proving the lemma? I suppose I'd need
> Cart_eq, which reads:
> 
> lemma Cart_eq: "(x = y) <--> (ALL i. x$i = y$i)"
> 
> but it quantifies over all i's. Since S1 and S2 have only 3
> dimensions, then should i be only in the range [1..3]?

The "2" in your type "real^2" is a type, not a member of the 
natural numbers type (or any other type).  You can write any 
integer, and if Isabelle believes it's of the type "2", it will be 
interpreted as a member of that type by taking its remainder 
modulo 2.  For example:

lemma "-6 = (2 :: 2)" by simp

So your r1 and r3 say exactly the same thing, but Isabelle didn't 
complain, because it can interpret them meaningfully.

So the quantified i in Cart_eq (when applied in your situation) is 
an element of the type "2", which has only two elements: "0" and 
"1" (or "1" and "2", if you prefer).  The trick is to turn this 
"ALL i." into a conjunction "x$1 = y$1 & x$2 = y$2", which can be 
quickly proven from your r2 and r3.  Fortunately, forall_2 can do 
just this, so:

lemma "S1 = S2"
  using r2 r3
  by (simp add: Cart_eq forall_2)

Multivariate_Analysis also has similar lemmas forall_1 and 
forall_3 for the types "1" and "3", but not forall_4 or greater.

I tested all this in Isabelle2009-2.  I haven't moved to 
Isabelle2011 yet, because my work relies heavily on what I've just 
described, and the following in the NEWS file for Isabelle2011 
worried me:

* Session Multivariate_Analysis: introduced a type class for 
euclidean space.  Most theorems are now stated in terms of 
euclidean spaces instead of finite cartesian products.

  types
    real ^ 'n ~>  'a::real_vector
              ~>  'a::euclidean_space
              ~>  'a::ordered_euclidean_space
        (depends on your needs)

  constants
     _ $ _        ~> _ $$ _
     \<chi> x. _  ~> \<chi>\<chi> x. _
     CARD('n)     ~> DIM('a)

Also note that the indices are now natural numbers and not from 
some finite type. Finite cartesian products of euclidean spaces, 
products of euclidean spaces the real and complex numbers are 
instantiated to be euclidean_spaces.  INCOMPATIBILITY.

> Thanks
> Steve

Tim
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