# Re: [isabelle] Using implications

`I agree with your proof. But then, if it is to define how to *approximate*
``a function f and a function g to be the same by comparing those x's and y's
``c, then should the = operator be replaced here? It seems the problem here
`

`is caused by overloading = with a form of approximated =, right? If so, how
``should one go about introducing a new operator, say for approximation?
`
Thank
Eg
On Jan 26, 2011 6:23pm, Brian Huffman <brianh at cs.pdx.edu> wrote:

On Wed, Jan 26, 2011 at 10:14 AM, Eg Gloor egglue at gmail.com> wrote:

> axiomatization

> S :: "real set" and

> foo :: "real => real" and

> bar :: "real => real" and

> bax :: "real => real" and

> c :: real

> where ax1: "ALL f g. (ALL x y. x > c & y > cfx = gy) --> f = g"

> and ax2: "ALL a b. bax a > c & bax b > c foo (bax a) = bar (bax b)"

Once again, we see that axioms are evil. Your ax1 is already

inconsistent all by itself:

lemma "False"

proof -

let ?f = "%x::real. if x > c then (0::real) else 1"

let ?g = "%x::real. if x > c then (0::real) else 2"

have "?f = ?g"

by (rule ax1 [rule_format], simp)

hence "?fc = ?g c"

by (rule arg_cong)

thus "False"

by simp

qed

-> Brian

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