The definition of suminf over the reals (in seqTheory) is completely different; is it clear that the extreal version can't mimic the original?
Michael On 18/9/19, 06:18, "Chun Tian (binghe)" <[email protected]> wrote: Hi, I occasionally found that HOL's current definition of ext_suminf (extrealTheory) has a bug: [ext_suminf_def] Definition ⊢ ∀f. suminf f = sup (IMAGE (λn. ∑ f (count n)) 𝕌(:num)) The purpose of `suminf f` is to calculate an infinite sum: f(0) + f(1) + .... To make the resulting summation "meaningful", all lemmas about ext_suminf assume that (!n. 0 <= f n) (f is positive). This also indeed how it's used in all applications. For instance, one lemma says, if f is positive, so is `suminf f`: [ext_suminf_pos] Theorem ⊢ ∀f. (∀n. 0 ≤ f n) ⇒ 0 ≤ suminf f However, I found that the above lemma can be proved even without knowing anything of `f`, see the following proofs: Theorem ext_suminf_pos_bug : !f. 0 <= ext_suminf f Proof RW_TAC std_ss [ext_suminf_def] >> MATCH_MP_TAC le_sup_imp' >> REWRITE_TAC [IN_IMAGE, IN_UNIV] >> Q.EXISTS_TAC `0` >> BETA_TAC >> REWRITE_TAC [COUNT_ZERO, EXTREAL_SUM_IMAGE_EMPTY] QED where [le_sup_imp'] is a version of [le_sup_imp] before applying IN_APP: [le_sup_imp'] Theorem ⊢ ∀p x. x ∈ p ⇒ x ≤ sup p For the reason, ext_suminf is actually calculating the sup of the following values: 0 f(0) f(0) + f(1) f(0) + f(1) + f(2) ... The first line (0) comes from `count 0 = {}` where `0 IN univ(:num)`, then SUM_IMAGE f {} = 0. Now consider f = (\n. -1), the above list of values are: 0, -1, -2, -3 .... But since the sup of a set containing 0 is at least 0, the `suminf f` in this case will be 0 (instead of the correct value -1). This is why `0 <= suminf f` holds for whatever f. I believe Isabelle's suminf (in Extended_Real.thy) has the same problem (but I can't find its definition, correct me if I'm wrong here), i.e. the following theorem holds even without the "assumes": lemma suminf_0_le: fixes f :: "nat ⇒ ereal" assumes "⋀n. 0 ≤ f n" shows "0 ≤ (∑n. f n)" using suminf_upper[of f 0, OF assms] by simp The solution is quite obvious. I'm going to fix HOL's definition of ext_suminf with the following one: val ext_suminf_def = Define `ext_suminf f = sup (IMAGE (\n. SIGMA f (count (SUC n))) UNIV)`; That is, using `SUC n` intead of `n` to eliminate the fake base case (0). I believe this change only causes minor incompatibilities. Any comment? Regards, Chun Tian _______________________________________________ hol-info mailing list [email protected] https://lists.sourceforge.net/lists/listinfo/hol-info
