- not_less
-
|- ¬(x < y) ⇔ y < x + 1
- elim_eq
-
|- (x = y) ⇔ x < y + 1 ∧ y < x + 1
- less_to_leq_samel
-
|- ∀x y. x < y ⇔ x ≤ y + -1
- less_to_leq_samer
-
|- ∀x y. x < y ⇔ x + 1 ≤ y
- lt_move_all_right
-
|- ∀x y. x < y ⇔ 0 < y + -x
- lt_move_all_left
-
|- ∀x y. x < y ⇔ x + -y < 0
- lt_move_left_left
-
|- ∀x y z. x < y + z ⇔ x + -y < z
- lt_move_left_right
-
|- ∀x y z. x + y < z ⇔ y < z + -x
- le_move_right_left
-
|- ∀x y z. x ≤ y + z ⇔ x + -z ≤ y
- le_move_all_right
-
|- ∀x y. x ≤ y ⇔ 0 ≤ y + -x
- eq_move_all_right
-
|- ∀x y. (x = y) ⇔ (0 = y + -x)
- eq_move_all_left
-
|- ∀x y. (x = y) ⇔ (x + -y = 0)
- eq_move_left_left
-
|- ∀x y z. (x = y + z) ⇔ (x + -y = z)
- eq_move_left_right
-
|- ∀x y z. (x + y = z) ⇔ (y = z + -x)
- eq_move_right_left
-
|- ∀x y z. (x = y + z) ⇔ (x + -z = y)
- lcm_eliminate
-
|- ∀P c. (∃x. P (c * x)) ⇔ ∃x. P x ∧ c int_divides x
- lt_justify_multiplication
-
|- ∀n x y. 0 < n ⇒ (x < y ⇔ n * x < n * y)
- eq_justify_multiplication
-
|- ∀n x y. 0 < n ⇒ ((x = y) ⇔ (n * x = n * y))
- justify_divides
-
|- ∀n x y. 0 < n ⇒ (x int_divides y ⇔ n * x int_divides n * y)
- justify_divides2
-
|- ∀n c x y.
n * x int_divides n * y + c ⇔
n * x int_divides n * y + c ∧ n int_divides c
- justify_divides3
-
|- ∀n x c. n int_divides n * x + c ⇔ n int_divides c
- INT_SUB_SUB3
-
|- ∀x y z. x − (y − z) = x + z − y
- move_sub
-
|- ∀c b a. a − c + b = a + b − c
- can_get_small
-
|- ∀x y d. 0 < d ⇒ ∃c. 0 < c ∧ y − c * d < x
- can_get_big
-
|- ∀x y d. 0 < d ⇒ ∃c. 0 < c ∧ x < y + c * d
- positive_product_implication
-
|- ∀c d. 0 < c ∧ 0 < d ⇒ 0 < c * d
- restricted_quantification_simp
-
|- ∀low high x.
low < x ∧ x ≤ high ⇔ low < high ∧ ((x = high) ∨ low < x ∧ x ≤ high − 1)
- top_and_lessers
-
|- ∀P d x0. (∀x. P x ⇒ P (x − d)) ∧ P x0 ⇒ ∀c. 0 < c ⇒ P (x0 − c * d)
- bot_and_greaters
-
|- ∀P d x0. (∀x. P x ⇒ P (x + d)) ∧ P x0 ⇒ ∀c. 0 < c ⇒ P (x0 + c * d)
- in_additive_range
-
|- ∀low d x. low < x ∧ x ≤ low + d ⇔ ∃j. (x = low + j) ∧ 0 < j ∧ j ≤ d
- in_subtractive_range
-
|- ∀high d x. high − d ≤ x ∧ x < high ⇔ ∃j. (x = high − j) ∧ 0 < j ∧ j ≤ d
- subtract_to_small
-
|- ∀x d. 0 < d ⇒ ∃k. 0 < x − k * d ∧ x − k * d ≤ d
- add_to_great
-
|- ∀x d. 0 < d ⇒ ∃k. 0 < x + k * d ∧ x + k * d ≤ d
- INT_LT_ADD_NUMERAL
-
|- ∀x y.
x < x + &NUMERAL (BIT1 y) ∧ x < x + &NUMERAL (BIT2 y) ∧
¬(x < x + -&NUMERAL y)
- INT_NUM_FORALL
-
|- (∀n. P (&n)) ⇔ ∀x. 0 ≤ x ⇒ P x
- INT_NUM_EXISTS
-
|- (∃n. P (&n)) ⇔ ∃x. 0 ≤ x ∧ P x
- INT_NUM_UEXISTS
-
|- (∃!n. P (&n)) ⇔ ∃!x. 0 ≤ x ∧ P x
- INT_NUM_SUB
-
|- ∀n m. &(n − m) = if &n < &m then 0 else &n − &m
- INT_NUM_COND
-
|- ∀b n m. &(if b then n else m) = if b then &n else &m
- INT_NUM_ODD
-
|- ∀n. ODD n ⇔ ¬(2 int_divides &n)
- INT_NUM_EVEN
-
|- ∀n. EVEN n ⇔ 2 int_divides &n
- HO_SUB_ELIM
-
|- ∀P a b. P (&(a − b)) ⇔ &b ≤ &a ∧ P (&a + -&b) ∨ &a < &b ∧ P 0
- CONJ_EQ_ELIM
-
|- ∀P v e. (v = e) ∧ P v ⇔ (v = e) ∧ P e
- elim_neg_ones
-
|- ∀x. x + -1 + 1 = x
- elim_minus_ones
-
|- ∀x. x + 1 − 1 = x
- INT_NUM_DIVIDES
-
|- ∀n m. &n int_divides &m ⇔ divides n m
- INT_LINEAR_GCD
-
|- ∀n m. ∃p q. p * &n + q * &m = &gcd n m
- INT_DIVIDES_LRMUL
-
|- ∀p q r. q ≠ 0 ⇒ (p * q int_divides r * q ⇔ p int_divides r)
- INT_DIVIDES_RELPRIME_MUL
-
|- ∀p q r. (gcd p q = 1) ⇒ (&p int_divides &q * r ⇔ &p int_divides r)
- gcdthm2
-
|- ∀m a x b d p q.
(d = gcd a m) ∧ (&d = p * &a + q * &m) ∧ d ≠ 0 ∧ m ≠ 0 ∧ a ≠ 0 ⇒
(&m int_divides &a * x + b ⇔
&d int_divides b ∧ ∃t. x = -p * (b / &d) + t * (&m / &d))
- gcd1thm
-
|- ∀m n p q. (p * &m + q * &n = 1) ⇒ (gcd m n = 1)
- gcd21_thm
-
|- ∀m a x b p q.
(p * &a + q * &m = 1) ∧ m ≠ 0 ∧ a ≠ 0 ⇒
(&m int_divides &a * x + b ⇔ ∃t. x = -p * b + t * &m)
- elim_lt_coeffs1
-
|- ∀n m x. m ≠ 0 ⇒ (&n < &m * x ⇔ &n / &m < x)
- elim_lt_coeffs2
-
|- ∀n m x.
m ≠ 0 ⇒
(&m * x < &n ⇔ x < if &m int_divides &n then &n / &m else &n / &m + 1)
- elim_le_coeffs
-
|- ∀m n x. 0 < m ⇒ (0 ≤ m * x + n ⇔ 0 ≤ x + n / m)
- elim_eq_coeffs
-
|- ∀m x y. m ≠ 0 ⇒ ((&m * x = y) ⇔ &m int_divides y ∧ (x = y / &m))
- cooper_lemma_1
-
|- ∀m n a b u v p q x d.
(d = gcd (u * m) (a * n)) ∧ (&d = p * &u * &m + q * &a * &n) ∧ m ≠ 0 ∧
n ≠ 0 ∧ a ≠ 0 ∧ u ≠ 0 ⇒
(&m int_divides &a * x + b ∧ &n int_divides &u * x + v ⇔
&m * &n int_divides &d * x + v * &m * p + b * &n * q ∧
&d int_divides &a * v − &u * b)
- bmarker_rewrites
-
|- ∀p q r.
(q ∧ bmarker p ⇔ bmarker p ∧ q) ∧
(q ∧ bmarker p ∧ r ⇔ bmarker p ∧ q ∧ r) ∧
((bmarker p ∧ q) ∧ r ⇔ bmarker p ∧ q ∧ r)
- NOT_INT_DIVIDES
-
|- ∀c d.
c ≠ 0 ⇒
(¬(c int_divides d) ⇔ ∃i. 1 ≤ i ∧ i ≤ ABS c − 1 ∧ c int_divides d + i)
- NOT_INT_DIVIDES_POS
-
|- ∀n d.
n ≠ 0 ⇒
(¬(&n int_divides d) ⇔ ∃i. (1 ≤ i ∧ i ≤ &n − 1) ∧ &n int_divides d + i)
- le_context_rwt1
-
|- 0 ≤ c + x ⇒ x ≤ y ⇒ (0 ≤ c + y ⇔ T)
- le_context_rwt2
-
|- 0 ≤ c + x ⇒ y < -x ⇒ (0 ≤ -c + y ⇔ F)
- le_context_rwt3
-
|- 0 ≤ c + x ⇒ x < y ⇒ ((0 = c + y) ⇔ F)
- le_context_rwt4
-
|- 0 ≤ c + x ⇒ x < -y ⇒ ((0 = -c + y) ⇔ F)
- le_context_rwt5
-
|- 0 ≤ c + x ⇒ (0 ≤ -c + -x ⇔ (0 = c + x))
- eq_context_rwt1
-
|- (0 = c + x) ⇒ (0 ≤ c + y ⇔ x ≤ y)
- eq_context_rwt2
-
|- (0 = c + x) ⇒ (0 ≤ -c + y ⇔ -x ≤ y)