- POLY_ADD_CLAUSES
-
|- ([] + p2 = p2) ∧ (p1 + [] = p1) ∧ ((h1::t1) + (h2::t2) = h1 + h2::t1 + t2)
- POLY_CMUL_CLAUSES
-
|- (c ## [] = []) ∧ (c ## (h::t) = c * h::c ## t)
- POLY_NEG_CLAUSES
-
|- (¬[] = []) ∧ (¬(h::t) = -h::¬t)
- POLY_MUL_CLAUSES
-
|- ([] * p2 = []) ∧ ([h1] * p2 = h1 ## p2) ∧
((h1::k1::t1) * p2 = h1 ## p2 + (0::(k1::t1) * p2))
- POLY_DIFF_CLAUSES
-
|- (diff [] = []) ∧ (diff [c] = []) ∧ (diff (h::t) = poly_diff_aux 1 t)
- POLY_ADD
-
|- ∀p1 p2 x. poly (p1 + p2) x = poly p1 x + poly p2 x
- POLY_CMUL
-
|- ∀p c x. poly (c ## p) x = c * poly p x
- POLY_NEG
-
|- ∀p x. poly (¬p) x = -poly p x
- POLY_MUL
-
|- ∀x p1 p2. poly (p1 * p2) x = poly p1 x * poly p2 x
- POLY_EXP
-
|- ∀p n x. poly (p poly_exp n) x = poly p x pow n
- POLY_DIFF_LEMMA
-
|- ∀l n x.
((λx. x pow SUC n * poly l x) diffl
(x pow n * poly (poly_diff_aux (SUC n) l) x)) x
- POLY_DIFF
-
|- ∀l x. ((λx. poly l x) diffl poly (diff l) x) x
- POLY_DIFFERENTIABLE
-
|- ∀l x. (λx. poly l x) differentiable x
- POLY_CONT
-
|- ∀l x. (λx. poly l x) contl x
- POLY_IVT_POS
-
|- ∀p a b.
a < b ∧ poly p a < 0 ∧ poly p b > 0 ⇒ ∃x. a < x ∧ x < b ∧ (poly p x = 0)
- POLY_IVT_NEG
-
|- ∀p a b.
a < b ∧ poly p a > 0 ∧ poly p b < 0 ⇒ ∃x. a < x ∧ x < b ∧ (poly p x = 0)
- POLY_MVT
-
|- ∀p a b.
a < b ⇒
∃x. a < x ∧ x < b ∧ (poly p b − poly p a = (b − a) * poly (diff p) x)
- POLY_ADD_RZERO
-
|- ∀p. poly (p + []) = poly p
- POLY_MUL_ASSOC
-
|- ∀p q r. poly (p * (q * r)) = poly (p * q * r)
- POLY_EXP_ADD
-
|- ∀d n p. poly (p poly_exp (n + d)) = poly (p poly_exp n * p poly_exp d)
- POLY_DIFF_AUX_ADD
-
|- ∀p1 p2 n.
poly (poly_diff_aux n (p1 + p2)) =
poly (poly_diff_aux n p1 + poly_diff_aux n p2)
- POLY_DIFF_AUX_CMUL
-
|- ∀p c n. poly (poly_diff_aux n (c ## p)) = poly (c ## poly_diff_aux n p)
- POLY_DIFF_AUX_NEG
-
|- ∀p n. poly (poly_diff_aux n (¬p)) = poly (¬poly_diff_aux n p)
- POLY_DIFF_AUX_MUL_LEMMA
-
|- ∀p n. poly (poly_diff_aux (SUC n) p) = poly (poly_diff_aux n p + p)
- POLY_DIFF_ADD
-
|- ∀p1 p2. poly (diff (p1 + p2)) = poly (diff p1 + diff p2)
- POLY_DIFF_CMUL
-
|- ∀p c. poly (diff (c ## p)) = poly (c ## diff p)
- POLY_DIFF_NEG
-
|- ∀p. poly (diff (¬p)) = poly (¬diff p)
- POLY_DIFF_MUL_LEMMA
-
|- ∀t h. poly (diff (h::t)) = poly ((0::diff t) + t)
- POLY_DIFF_MUL
-
|- ∀p1 p2. poly (diff (p1 * p2)) = poly (p1 * diff p2 + diff p1 * p2)
- POLY_DIFF_EXP
-
|- ∀p n.
poly (diff (p poly_exp SUC n)) = poly (&SUC n ## p poly_exp n * diff p)
- POLY_DIFF_EXP_PRIME
-
|- ∀n a.
poly (diff ([-a; 1] poly_exp SUC n)) =
poly (&SUC n ## [-a; 1] poly_exp n)
- POLY_LINEAR_REM
-
|- ∀t h. ∃q r. h::t = [r] + [-a; 1] * q
- POLY_LINEAR_DIVIDES
-
|- ∀a p. (poly p a = 0) ⇔ (p = []) ∨ ∃q. p = [-a; 1] * q
- POLY_LENGTH_MUL
-
|- ∀q. LENGTH ([-a; 1] * q) = SUC (LENGTH q)
- POLY_ROOTS_INDEX_LEMMA
-
|- ∀n p.
poly p ≠ poly [] ∧ (LENGTH p = n) ⇒
∃i. ∀x. (poly p x = 0) ⇒ ∃m. m ≤ n ∧ (x = i m)
- POLY_ROOTS_INDEX_LENGTH
-
|- ∀p.
poly p ≠ poly [] ⇒ ∃i. ∀x. (poly p x = 0) ⇒ ∃n. n ≤ LENGTH p ∧ (x = i n)
- POLY_ROOTS_FINITE_LEMMA
-
|- ∀p. poly p ≠ poly [] ⇒ ∃N i. ∀x. (poly p x = 0) ⇒ ∃n. n < N ∧ (x = i n)
- FINITE_LEMMA
-
|- ∀i N P. (∀x. P x ⇒ ∃n. n < N ∧ (x = i n)) ⇒ ∃a. ∀x. P x ⇒ x < a
- POLY_ROOTS_FINITE
-
|- ∀p. poly p ≠ poly [] ⇔ ∃N i. ∀x. (poly p x = 0) ⇒ ∃n. n < N ∧ (x = i n)
- POLY_ENTIRE_LEMMA
-
|- ∀p q. poly p ≠ poly [] ∧ poly q ≠ poly [] ⇒ poly (p * q) ≠ poly []
- POLY_ENTIRE
-
|- ∀p q. (poly (p * q) = poly []) ⇔ (poly p = poly []) ∨ (poly q = poly [])
- POLY_MUL_LCANCEL
-
|- ∀p q r.
(poly (p * q) = poly (p * r)) ⇔ (poly p = poly []) ∨ (poly q = poly r)
- POLY_EXP_EQ_0
-
|- ∀p n. (poly (p poly_exp n) = poly []) ⇔ (poly p = poly []) ∧ n ≠ 0
- POLY_PRIME_EQ_0
-
|- ∀a. poly [a; 1] ≠ poly []
- POLY_EXP_PRIME_EQ_0
-
|- ∀a n. poly ([a; 1] poly_exp n) ≠ poly []
- POLY_ZERO_LEMMA
-
|- ∀h t. (poly (h::t) = poly []) ⇒ (h = 0) ∧ (poly t = poly [])
- POLY_ZERO
-
|- ∀p. (poly p = poly []) ⇔ EVERY (λc. c = 0) p
- POLY_DIFF_AUX_ISZERO
-
|- ∀p n. EVERY (λc. c = 0) (poly_diff_aux (SUC n) p) ⇔ EVERY (λc. c = 0) p
- POLY_DIFF_ISZERO
-
|- ∀p. (poly (diff p) = poly []) ⇒ ∃h. poly p = poly [h]
- POLY_DIFF_ZERO
-
|- ∀p. (poly p = poly []) ⇒ (poly (diff p) = poly [])
- POLY_DIFF_WELLDEF
-
|- ∀p q. (poly p = poly q) ⇒ (poly (diff p) = poly (diff q))
- POLY_PRIMES
-
|- ∀a p q.
[a; 1] poly_divides p * q ⇔ [a; 1] poly_divides p ∨ [a; 1] poly_divides q
- POLY_DIVIDES_REFL
-
|- ∀p. p poly_divides p
- POLY_DIVIDES_TRANS
-
|- ∀p q r. p poly_divides q ∧ q poly_divides r ⇒ p poly_divides r
- POLY_DIVIDES_EXP
-
|- ∀p m n. m ≤ n ⇒ p poly_exp m poly_divides p poly_exp n
- POLY_EXP_DIVIDES
-
|- ∀p q m n. p poly_exp n poly_divides q ∧ m ≤ n ⇒ p poly_exp m poly_divides q
- POLY_DIVIDES_ADD
-
|- ∀p q r. p poly_divides q ∧ p poly_divides r ⇒ p poly_divides q + r
- POLY_DIVIDES_SUB
-
|- ∀p q r. p poly_divides q ∧ p poly_divides q + r ⇒ p poly_divides r
- POLY_DIVIDES_SUB2
-
|- ∀p q r. p poly_divides r ∧ p poly_divides q + r ⇒ p poly_divides q
- POLY_DIVIDES_ZERO
-
|- ∀p q. (poly p = poly []) ⇒ q poly_divides p
- POLY_ORDER_EXISTS
-
|- ∀a d p.
(LENGTH p = d) ∧ poly p ≠ poly [] ⇒
∃n.
[-a; 1] poly_exp n poly_divides p ∧
¬([-a; 1] poly_exp SUC n poly_divides p)
- POLY_ORDER
-
|- ∀p a.
poly p ≠ poly [] ⇒
∃!n.
[-a; 1] poly_exp n poly_divides p ∧
¬([-a; 1] poly_exp SUC n poly_divides p)
- ORDER
-
|- ∀p a n.
[-a; 1] poly_exp n poly_divides p ∧
¬([-a; 1] poly_exp SUC n poly_divides p) ⇔
(n = poly_order a p) ∧ poly p ≠ poly []
- ORDER_THM
-
|- ∀p a.
poly p ≠ poly [] ⇒
[-a; 1] poly_exp poly_order a p poly_divides p ∧
¬([-a; 1] poly_exp SUC (poly_order a p) poly_divides p)
- ORDER_UNIQUE
-
|- ∀p a n.
poly p ≠ poly [] ∧ [-a; 1] poly_exp n poly_divides p ∧
¬([-a; 1] poly_exp SUC n poly_divides p) ⇒
(n = poly_order a p)
- ORDER_POLY
-
|- ∀p q a. (poly p = poly q) ⇒ (poly_order a p = poly_order a q)
- ORDER_ROOT
-
|- ∀p a. (poly p a = 0) ⇔ (poly p = poly []) ∨ poly_order a p ≠ 0
- ORDER_DIVIDES
-
|- ∀p a n.
[-a; 1] poly_exp n poly_divides p ⇔
(poly p = poly []) ∨ n ≤ poly_order a p
- ORDER_DECOMP
-
|- ∀p a.
poly p ≠ poly [] ⇒
∃q.
(poly p = poly ([-a; 1] poly_exp poly_order a p * q)) ∧
¬([-a; 1] poly_divides q)
- ORDER_MUL
-
|- ∀a p q.
poly (p * q) ≠ poly [] ⇒
(poly_order a (p * q) = poly_order a p + poly_order a q)
- ORDER_DIFF
-
|- ∀p a.
poly (diff p) ≠ poly [] ∧ poly_order a p ≠ 0 ⇒
(poly_order a p = SUC (poly_order a (diff p)))
- POLY_SQUAREFREE_DECOMP_ORDER
-
|- ∀p q d e r s.
poly (diff p) ≠ poly [] ∧ (poly p = poly (q * d)) ∧
(poly (diff p) = poly (e * d)) ∧ (poly d = poly (r * p + s * diff p)) ⇒
∀a. poly_order a q = if poly_order a p = 0 then 0 else 1
- RSQUAREFREE_ROOTS
-
|- ∀p. rsquarefree p ⇔ ∀a. ¬((poly p a = 0) ∧ (poly (diff p) a = 0))
- RSQUAREFREE_DECOMP
-
|- ∀p a.
rsquarefree p ∧ (poly p a = 0) ⇒
∃q. (poly p = poly ([-a; 1] * q)) ∧ poly q a ≠ 0
- POLY_SQUAREFREE_DECOMP
-
|- ∀p q d e r s.
poly (diff p) ≠ poly [] ∧ (poly p = poly (q * d)) ∧
(poly (diff p) = poly (e * d)) ∧ (poly d = poly (r * p + s * diff p)) ⇒
rsquarefree q ∧ ∀a. (poly q a = 0) ⇔ (poly p a = 0)
- POLY_NORMALIZE
-
|- ∀p. poly (normalize p) = poly p
- DEGREE_ZERO
-
|- ∀p. (poly p = poly []) ⇒ (degree p = 0)
- POLY_ROOTS_FINITE_SET
-
|- ∀p. poly p ≠ poly [] ⇒ FINITE {x | poly p x = 0}
- POLY_MONO
-
|- ∀x k p. abs x ≤ k ⇒ abs (poly p x) ≤ poly (MAP abs p) k