The Gauge Integral Theory in HOL4

Table 2

The lemmas proving Theorem 19.

Name of lemma	Description in HOL4
DIVISION_LE_SUC	∀ 𝑑𝑎𝑏. division (a, b) d ==> ∀n. d n <= d (SUC n)
DIVISION_MONO_LE	∀𝑑𝑎𝑏. division (a, b) d ==> ∀m n. m <= n ==> d m <= d n
DIVISION_MONO_LE_SUC	∀𝑑𝑎𝑏. division (a, b) d ==> ∀n. d n <= d (SUC n)
DIVISION_INTERMEDIATE	∀ 𝑑𝑎𝑏𝑐. division (a, b) d  ∧ a <= c  ∧ c <= b ==>; ∃ n. n <= dsize d  ∧ d n <= c  ∧ c <= d (SUC n)
DIVISION_DSIZE_LE	∀ 𝑎𝑏𝑑𝑛. division (a, b) d ∧ (d (SUC n) = d n) ==> dsize d <= n
DIVISION_DSIZE_GE	∀ 𝑎𝑏𝑑𝑛. division (a, b) d  ∧ d n < d (SUC n) ==> SUC n <= dsize d
DIVISION_DSIZE_EQ	∀ 𝑎𝑏𝑑𝑛. division (a, b) d  ∧ d n < d (SUC n) ∧ (d (SUC (SUC n)) = d (SUC n)) ==>; (dsize d = SUC n)
DIVISION_DSIZE_EQ_ALT	∀ 𝑎𝑏𝑑𝑛. division (a, b) d ∧ (d (SUC n) = d n) ∧ (∀ i. i < n ==> d i < d (SUC i)); ==> (dsize d = n)