Theory Cring_Poly
theory Cring_Poly
imports "HOL-Algebra.UnivPoly" "HOL-Algebra.Subrings" Function_Ring
begin
text‹
This theory extends the material in \<^theory>‹HOL-Algebra.UnivPoly›. The main additions are
material on Taylor expansions of polynomials and polynomial derivatives, and various applications
of the universal property of polynomial evaluation. These include construing polynomials as
functions from the base ring to itself, composing one polynomial with another, and extending
homomorphisms between rings to homomoprhisms of their polynomial rings. These formalizations
are necessary components of the proof of Hensel's lemma for $p$-adic integers, and for the
proof of $p$-adic quantifier elimination. ›
lemma(in ring) ring_hom_finsum:
assumes "h ∈ ring_hom R S"
assumes "ring S"
assumes "finite I"
assumes "F ∈ I → carrier R"
shows "h (finsum R F I) = finsum S (h ∘ F) I"
proof-
have I: "(h ∈ ring_hom R S ∧ F ∈ I → carrier R) ⟶ h (finsum R F I) = finsum S (h ∘ F) I"
apply(rule finite_induct, rule assms)
using assms ring_hom_zero[of h R S]
apply (metis abelian_group_def abelian_monoid.finsum_empty ring_axioms ring_def)
proof(rule)
fix A a
assume A: "finite A" "a ∉ A" "h ∈ ring_hom R S ∧ F ∈ A → carrier R ⟶
h (finsum R F A) = finsum S (h ∘ F) A" "h ∈ ring_hom R S ∧ F ∈ insert a A → carrier R"
have 0: "h ∈ ring_hom R S ∧ F ∈ A → carrier R "
using A by auto
have 1: "h (finsum R F A) = finsum S (h ∘ F) A"
using A 0 by auto
have 2: "abelian_monoid S"
using assms ring_def abelian_group_def by auto
have 3: "h (F a ⊕ finsum R F A) = h (F a) ⊕⇘S⇙ (finsum S (h ∘ F) A) "
using ring_hom_add assms finsum_closed 1 A(4) by fastforce
have 4: "finsum R F (insert a A) = F a ⊕ finsum R F A"
using finsum_insert[of A a F] A assms by auto
have 5: "finsum S (h ∘ F) (insert a A) = (h ∘ F) a ⊕⇘S⇙ finsum S (h ∘ F) A"
apply(rule abelian_monoid.finsum_insert[of S A a "h ∘ F"])
apply (simp add: "2")
apply(rule A)
apply(rule A)
using ring_hom_closed A "0" apply fastforce
using A ring_hom_closed by auto
show "h (finsum R F (insert a A)) =
finsum S (h ∘ F) (insert a A)"
unfolding 4 5 3 by auto
qed
thus ?thesis using assms by blast
qed
lemma(in ring) ring_hom_a_inv:
assumes "ring S"
assumes "h ∈ ring_hom R S"
assumes "b ∈ carrier R"
shows "h (⊖ b) = ⊖⇘S⇙ h b"
proof-
have "h b ⊕⇘S⇙ h (⊖ b) = 𝟬⇘S⇙"
by (metis (no_types, opaque_lifting) abelian_group.a_inv_closed assms(1) assms(2) assms(3)
is_abelian_group local.ring_axioms r_neg ring_hom_add ring_hom_zero)
then show ?thesis
by (metis (no_types, lifting) abelian_group.minus_equality add.inv_closed assms(1)
assms(2) assms(3) ring.is_abelian_group ring.ring_simprules(10) ring_hom_closed)
qed
lemma(in ring) ring_hom_minus:
assumes "ring S"
assumes "h ∈ ring_hom R S"
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "h (a ⊖ b) = h a ⊖⇘S⇙ h b"
using assms ring_hom_add[of h R S a "⊖⇘R⇙ b"]
unfolding a_minus_def
using ring_hom_a_inv[of S h b] by auto
lemma ring_hom_nat_pow:
assumes "ring R"
assumes "ring S"
assumes "h ∈ ring_hom R S"
assumes "a ∈ carrier R"
shows "h (a[^]⇘R⇙(n::nat)) = (h a)[^]⇘S⇙(n::nat)"
using assms by (simp add: ring_hom_ring.hom_nat_pow ring_hom_ringI2)
lemma (in ring) Units_not_right_zero_divisor:
assumes "a ∈ Units R"
assumes "b ∈ carrier R"
assumes "a ⊗ b = 𝟬"
shows "b = 𝟬"
proof-
have "inv a ⊗ a ⊗ b = 𝟬 "
using assms Units_closed Units_inv_closed r_null m_assoc[of "inv a" a b] by presburger
thus ?thesis using assms
by (metis Units_l_inv l_one)
qed
lemma (in ring) Units_not_left_zero_divisor:
assumes "a ∈ Units R"
assumes "b ∈ carrier R"
assumes "b ⊗ a = 𝟬"
shows "b = 𝟬"
proof-
have "b ⊗ (a ⊗ inv a) = 𝟬 "
using assms Units_closed Units_inv_closed l_null m_assoc[of b a"inv a"] by presburger
thus ?thesis using assms
by (metis Units_r_inv r_one)
qed
lemma (in cring) finsum_remove:
assumes "⋀i. i ∈ Y ⟹ f i ∈ carrier R"
assumes "finite Y"
assumes "i ∈ Y"
shows "finsum R f Y = f i ⊕ finsum R f (Y - {i})"
proof-
have "finsum R f (insert i (Y - {i})) = f i ⊕ finsum R f (Y - {i})"
apply(rule finsum_insert)
using assms apply blast apply blast using assms apply blast
using assms by blast
thus ?thesis using assms
by (metis insert_Diff)
qed
type_synonym degree = nat
text‹The composition of two ring homomorphisms is a ring homomorphism›
lemma ring_hom_compose:
assumes "ring R"
assumes "ring S"
assumes "ring T"
assumes "h ∈ ring_hom R S"
assumes "g ∈ ring_hom S T"
assumes "⋀c. c ∈ carrier R ⟹ f c = g (h c)"
shows "f ∈ ring_hom R T"
proof(rule ring_hom_memI)
show "⋀x. x ∈ carrier R ⟹ f x ∈ carrier T"
using assms by (metis ring_hom_closed)
show " ⋀x y. x ∈ carrier R ⟹ y ∈ carrier R ⟹ f (x ⊗⇘R⇙ y) = f x ⊗⇘T⇙ f y"
proof-
fix x y
assume A: "x ∈ carrier R" "y ∈ carrier R"
show "f (x ⊗⇘R⇙ y) = f x ⊗⇘T⇙ f y"
proof-
have "f (x ⊗⇘R⇙ y) = g (h (x ⊗⇘R⇙ y))"
by (simp add: A(1) A(2) assms(1) assms(6) ring.ring_simprules(5))
then have "f (x ⊗⇘R⇙ y) = g ((h x) ⊗⇘S⇙ (h y))"
using A(1) A(2) assms(4) ring_hom_mult by fastforce
then have "f (x ⊗⇘R⇙ y) = g (h x) ⊗⇘T⇙ g (h y)"
using A(1) A(2) assms(4) assms(5) ring_hom_closed ring_hom_mult by fastforce
then show ?thesis
by (simp add: A(1) A(2) assms(6))
qed
qed
show "⋀x y. x ∈ carrier R ⟹ y ∈ carrier R ⟹ f (x ⊕⇘R⇙ y) = f x ⊕⇘T⇙ f y"
proof-
fix x y
assume A: "x ∈ carrier R" "y ∈ carrier R"
show "f (x ⊕⇘R⇙ y) = f x ⊕⇘T⇙ f y"
proof-
have "f (x ⊕⇘R⇙ y) = g (h (x ⊕⇘R⇙ y))"
by (simp add: A(1) A(2) assms(1) assms(6) ring.ring_simprules(1))
then have "f (x ⊕⇘R⇙ y) = g ((h x) ⊕⇘S⇙ (h y))"
using A(1) A(2) assms(4) ring_hom_add by fastforce
then have "f (x ⊕⇘R⇙ y) = g (h x) ⊕⇘T⇙ g (h y)"
by (metis (no_types, opaque_lifting) A(1) A(2) assms(4) assms(5) ring_hom_add ring_hom_closed)
then show ?thesis
by (simp add: A(1) A(2) assms(6))
qed
qed
show "f 𝟭⇘R⇙ = 𝟭⇘T⇙"
by (metis assms(1) assms(4) assms(5) assms(6) ring.ring_simprules(6) ring_hom_one)
qed
section‹Basic Notions about Polynomials›
context UP_ring
begin
text‹rings are closed under monomial terms›
lemma monom_term_car:
assumes "c ∈ carrier R"
assumes "x ∈ carrier R"
shows "c ⊗ x[^](n::nat) ∈ carrier R"
using assms monoid.nat_pow_closed
by blast
text‹Univariate polynomial ring over R›
lemma P_is_UP_ring:
"UP_ring R"
by (simp add: UP_ring_axioms)
text‹Degree function›
abbreviation(input) degree where
"degree f ≡ deg R f"
lemma UP_car_memI:
assumes "⋀n. n > k ⟹ p n = 𝟬"
assumes "⋀n. p n ∈ carrier R"
shows "p ∈ carrier P"
proof-
have "bound 𝟬 k p"
by (simp add: assms(1) bound.intro)
then show ?thesis
by (metis (no_types, lifting) P_def UP_def assms(2) mem_upI partial_object.select_convs(1))
qed
lemma(in UP_cring) UP_car_memI':
assumes "⋀x. g x ∈ carrier R"
assumes "⋀x. x > k ⟹ g x = 𝟬"
shows "g ∈ carrier (UP R)"
proof-
have "bound 𝟬 k g"
using assms unfolding bound_def by blast
then show ?thesis
using P_def UP_car_memI assms(1) by blast
qed
lemma(in UP_cring) UP_car_memE:
assumes "g ∈ carrier (UP R)"
shows "⋀x. g x ∈ carrier R"
"⋀x. x > (deg R g) ⟹ g x = 𝟬"
using P_def assms UP_def[of R] apply (simp add: mem_upD)
using assms UP_def[of R] up_def[of R]
by (smt (verit, del_insts) UP_ring.deg_aboveD is_UP_ring partial_object.select_convs(1) restrict_apply up_ring.select_convs(2))
end
subsection‹Lemmas About Coefficients›
context UP_ring
begin
text‹The goal here is to reduce dependence on the function coeff from Univ\_Poly, in favour of using
a polynomial itself as its coefficient function.›
lemma coeff_simp:
assumes "f ∈ carrier P"
shows "coeff (UP R) f = f "
proof fix x show "coeff (UP R) f x = f x"
using assms P_def UP_def[of R] by auto
qed
text‹Coefficients are in R›
lemma cfs_closed:
assumes "f ∈ carrier P"
shows "f n ∈ carrier R"
using assms coeff_simp[of f] P_def coeff_closed
by fastforce
lemma cfs_monom:
"a ∈ carrier R ==> (monom P a m) n = (if m=n then a else 𝟬)"
using coeff_simp P_def coeff_monom monom_closed by auto
lemma cfs_zero [simp]: "𝟬⇘P⇙ n = 𝟬"
using P_def UP_zero_closed coeff_simp coeff_zero by auto
lemma cfs_one [simp]: "𝟭⇘P⇙ n = (if n=0 then 𝟭 else 𝟬)"
by (metis P_def R.one_closed UP_ring.cfs_monom UP_ring_axioms monom_one)
lemma cfs_smult [simp]:
"[| a ∈ carrier R; p ∈ carrier P |] ==> (a ⊙⇘P⇙ p) n = a ⊗ p n"
using P_def UP_ring.coeff_simp UP_ring_axioms UP_smult_closed coeff_smult by fastforce
lemma cfs_add [simp]:
"[| p ∈ carrier P; q ∈ carrier P |] ==> (p ⊕⇘P⇙ q) n = p n ⊕ q n"
by (metis P.add.m_closed P_def UP_ring.coeff_add UP_ring.coeff_simp UP_ring_axioms)
lemma cfs_a_inv [simp]:
assumes R: "p ∈ carrier P"
shows "(⊖⇘P⇙ p) n = ⊖ (p n)"
using P.add.inv_closed P_def UP_ring.coeff_a_inv UP_ring.coeff_simp UP_ring_axioms assms
by fastforce
lemma cfs_minus [simp]:
"[| p ∈ carrier P; q ∈ carrier P |] ==> (p ⊖⇘P⇙ q) n = p n ⊖ q n"
using P.minus_closed P_def coeff_minus coeff_simp by auto
lemma cfs_monom_mult_r:
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
shows "(monom P a n ⊗⇘P⇙ p) (k + n) = a ⊗ p k"
using coeff_monom_mult assms P.m_closed P_def coeff_simp monom_closed by auto
lemma(in UP_cring) cfs_monom_mult_l:
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
shows "(p ⊗⇘P⇙ monom P a n) (k + n) = a ⊗ p k"
using UP_m_comm assms(1) assms(2) cfs_monom_mult_r by auto
lemma(in UP_cring) cfs_monom_mult_l':
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
assumes "m ≥ n"
shows "(f ⊗⇘P⇙ (monom P a n)) m = a ⊗ (f (m - n))"
using cfs_monom_mult_l[of f a n "m-n"] assms
by simp
lemma(in UP_cring) cfs_monom_mult_r':
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
assumes "m ≥ n"
shows "((monom P a n) ⊗⇘P⇙ f) m = a ⊗ (f (m - n))"
using cfs_monom_mult_r[of f a n "m-n"] assms
by simp
end
subsection‹Degree Bound Lemmas›
context UP_ring
begin
lemma bound_deg_sum:
assumes " f ∈ carrier P"
assumes "g ∈ carrier P"
assumes "degree f ≤ n"
assumes "degree g ≤ n"
shows "degree (f ⊕⇘P⇙ g) ≤ n"
using P_def UP_ring_axioms assms(1) assms(2) assms(3) assms(4)
by (meson deg_add max.boundedI order_trans)
lemma bound_deg_sum':
assumes " f ∈ carrier P"
assumes "g ∈ carrier P"
assumes "degree f < n"
assumes "degree g < n"
shows "degree (f ⊕⇘P⇙ g) < n"
using P_def UP_ring_axioms assms(1) assms(2)
assms(3) assms(4)
by (metis bound_deg_sum le_neq_implies_less less_imp_le_nat not_less)
lemma equal_deg_sum:
assumes " f ∈ carrier P"
assumes "g ∈ carrier P"
assumes "degree f < n"
assumes "degree g = n"
shows "degree (f ⊕⇘P⇙ g) = n"
proof-
have 0: "degree (f ⊕⇘P⇙ g) ≤n"
using assms bound_deg_sum
P_def UP_ring_axioms by auto
show "degree (f ⊕⇘P⇙ g) = n"
proof(rule ccontr)
assume "degree (f ⊕⇘P⇙ g) ≠ n "
then have 1: "degree (f ⊕⇘P⇙ g) < n"
using 0 by auto
have 2: "degree (⊖⇘P⇙ f) < n"
using assms by simp
have 3: "g = (f ⊕⇘P⇙ g) ⊕⇘P⇙ (⊖⇘P⇙ f)"
using assms
by (simp add: P.add.m_comm P.r_neg1)
then show False using 1 2 3 assms
by (metis UP_a_closed UP_a_inv_closed deg_add leD le_max_iff_disj)
qed
qed
lemma equal_deg_sum':
assumes "f ∈ carrier P"
assumes "g ∈ carrier P"
assumes "degree g < n"
assumes "degree f = n"
shows "degree (f ⊕⇘P⇙ g) = n"
using P_def UP_a_comm UP_ring.equal_deg_sum UP_ring_axioms assms(1) assms(2) assms(3) assms(4)
by fastforce
lemma degree_of_sum_diff_degree:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "degree q < degree p"
shows "degree (p ⊕⇘P⇙ q) = degree p"
by(rule equal_deg_sum', auto simp: assms)
lemma degree_of_difference_diff_degree:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "degree q < degree p"
shows "degree (p ⊖⇘P⇙ q) = degree p"
proof-
have A: "(p ⊖⇘P⇙ q) = p ⊕⇘P⇙ (⊖⇘P⇙ q)"
by (simp add: P.minus_eq)
have "degree (⊖⇘P⇙ q) = degree q "
by (simp add: assms(2))
then show ?thesis
using assms A
by (simp add: degree_of_sum_diff_degree)
qed
lemma (in UP_ring) deg_diff_by_const:
assumes "g ∈ carrier (UP R)"
assumes "a ∈ carrier R"
assumes "h = g ⊕⇘UP R⇙ up_ring.monom (UP R) a 0"
shows "deg R g = deg R h"
unfolding assms using assms
by (metis P_def UP_ring.bound_deg_sum UP_ring.deg_monom_le UP_ring.monom_closed UP_ring_axioms degree_of_sum_diff_degree gr_zeroI not_less)
lemma (in UP_ring) deg_diff_by_const':
assumes "g ∈ carrier (UP R)"
assumes "a ∈ carrier R"
assumes "h = g ⊖⇘UP R⇙ up_ring.monom (UP R) a 0"
shows "deg R g = deg R h"
apply(rule deg_diff_by_const[of _ "⊖ a"])
using assms apply blast
using assms apply blast
by (metis P.minus_eq P_def assms(2) assms(3) monom_a_inv)
lemma(in UP_ring) deg_gtE:
assumes "p ∈ carrier P"
assumes "i > deg R p"
shows "p i = 𝟬"
using assms P_def coeff_simp deg_aboveD by metis
end
subsection‹Leading Term Function›
definition leading_term where
"leading_term R f = monom (UP R) (f (deg R f)) (deg R f)"
context UP_ring
begin
abbreviation(input) ltrm where
"ltrm f ≡ monom P (f (deg R f)) (deg R f)"
text‹leading term is a polynomial›
lemma ltrm_closed:
assumes "f ∈ carrier P"
shows "ltrm f ∈ carrier P"
using assms
by (simp add: cfs_closed)
text‹Simplified coefficient function description for leading term›
lemma ltrm_coeff:
assumes "f ∈ carrier P"
shows "coeff P (ltrm f) n = (if (n = degree f) then (f (degree f)) else 𝟬)"
using assms
by (simp add: cfs_closed)
lemma ltrm_cfs:
assumes "f ∈ carrier P"
shows "(ltrm f) n = (if (n = degree f) then (f (degree f)) else 𝟬)"
using assms
by (simp add: cfs_closed cfs_monom)
lemma ltrm_cfs_above_deg:
assumes "f ∈ carrier P"
assumes "n > degree f"
shows "ltrm f n = 𝟬"
using assms
by (simp add: ltrm_cfs)
text‹The leading term of f has the same degree as f›
lemma deg_ltrm:
assumes "f ∈ carrier P"
shows "degree (ltrm f) = degree f"
using assms
by (metis P_def UP_ring.lcoeff_nonzero_deg UP_ring_axioms cfs_closed coeff_simp deg_const deg_monom)
text‹Subtracting the leading term yields a drop in degree›
lemma minus_ltrm_degree_drop:
assumes "f ∈ carrier P"
assumes "degree f = Suc n"
shows "degree (f ⊖⇘P⇙ (ltrm f)) ≤ n"
proof(rule UP_ring.deg_aboveI)
show C0: "UP_ring R"
by (simp add: UP_ring_axioms)
show C1: "f ⊖⇘P⇙ ltrm f ∈ carrier (UP R)"
using assms ltrm_closed P.minus_closed P_def
by blast
show C2: "⋀m. n < m ⟹ coeff (UP R) (f ⊖⇘P⇙ ltrm f) m = 𝟬"
proof-
fix m
assume A: "n<m"
show "coeff (UP R) (f ⊖⇘P⇙ ltrm f) m = 𝟬"
proof(cases " m = Suc n")
case True
have B: "f m ∈ carrier R"
using UP.coeff_closed P_def assms(1) cfs_closed by blast
have "m = degree f"
using True by (simp add: assms(2))
then have "f m = (ltrm f) m"
using ltrm_cfs assms(1) by auto
then have "(f m) ⊖⇘R⇙( ltrm f) m = 𝟬"
using B UP_ring_def P_is_UP_ring
B R.add.r_inv R.is_abelian_group abelian_group.minus_eq by fastforce
then have "(f ⊖⇘UP R⇙ ltrm f) m = 𝟬"
by (metis C1 ltrm_closed P_def assms(1) coeff_minus coeff_simp)
then show ?thesis
using C1 P_def UP_ring.coeff_simp UP_ring_axioms by fastforce
next
case False
have D0: "m > degree f" using False
using A assms(2) by linarith
have B: "f m ∈ carrier R"
using UP.coeff_closed P_def assms(1) cfs_closed
by blast
have "f m = (ltrm f) m"
using D0 ltrm_cfs_above_deg P_def assms(1) coeff_simp deg_aboveD
by auto
then show ?thesis
by (metis B ltrm_closed P_def R.r_neg UP_ring.coeff_simp UP_ring_axioms a_minus_def assms(1) coeff_minus)
qed
qed
qed
lemma ltrm_decomp:
assumes "f ∈ carrier P"
assumes "degree f >(0::nat)"
obtains g where "g ∈ carrier P ∧ f = g ⊕⇘P⇙ (ltrm f) ∧ degree g < degree f"
proof-
have 0: "f ⊖⇘P⇙ (ltrm f) ∈ carrier P"
using ltrm_closed assms(1) by blast
have 1: "f = (f ⊖⇘P⇙ (ltrm f)) ⊕⇘P⇙ (ltrm f)"
using assms
by (metis "0" ltrm_closed P.add.inv_solve_right P.minus_eq)
show ?thesis using assms 0 1 minus_ltrm_degree_drop[of f]
by (metis ltrm_closed Suc_diff_1 Suc_n_not_le_n deg_ltrm equal_deg_sum' linorder_neqE_nat that)
qed
text‹leading term of a sum›
lemma coeff_of_sum_diff_degree0:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "degree q < n"
shows "(p ⊕⇘P⇙ q) n = p n"
using assms P_def UP_ring.deg_aboveD UP_ring_axioms cfs_add coeff_simp cfs_closed deg_aboveD
by auto
lemma coeff_of_sum_diff_degree1:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "degree q < degree p"
shows "(p ⊕⇘P⇙ q) (degree p) = p (degree p)"
using assms(1) assms(2) assms(3) coeff_of_sum_diff_degree0 by blast
lemma ltrm_of_sum_diff_degree:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "degree p > degree q"
shows "ltrm (p ⊕⇘P⇙ q) = ltrm p"
unfolding leading_term_def
using assms(1) assms(2) assms(3) coeff_of_sum_diff_degree1 degree_of_sum_diff_degree
by presburger
text‹leading term of a monomial›
lemma ltrm_monom:
assumes "a ∈ carrier R"
assumes "f = monom P a n"
shows "ltrm f = f"
unfolding leading_term_def
by (metis P_def UP_ring.cfs_monom UP_ring.monom_zero UP_ring_axioms assms(1) assms(2) deg_monom)
lemma ltrm_monom_simp:
assumes "a ∈ carrier R"
shows "ltrm (monom P a n) = monom P a n"
using assms ltrm_monom by auto
lemma ltrm_inv_simp[simp]:
assumes "f ∈ carrier P"
shows "ltrm (ltrm f) = ltrm f"
by (metis assms deg_ltrm ltrm_cfs)
lemma ltrm_deg_0:
assumes "p ∈ carrier P"
assumes "degree p = 0"
shows "ltrm p = p"
using ltrm_monom assms P_def UP_ring.deg_zero_impl_monom UP_ring_axioms coeff_simp
by fastforce
lemma ltrm_prod_ltrm:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
shows "ltrm ((ltrm p) ⊗⇘P⇙ (ltrm q)) = (ltrm p) ⊗⇘P⇙ (ltrm q)"
using ltrm_monom R.m_closed assms(1) assms(2) cfs_closed monom_mult
by metis
text‹lead coefficient function›
abbreviation(input) lcf where
"lcf p ≡ p (deg R p)"
lemma(in UP_ring) lcf_ltrm:
"ltrm p = monom P (lcf p) (degree p)"
by auto
lemma lcf_closed:
assumes "f ∈ carrier P"
shows "lcf f ∈ carrier R"
by (simp add: assms cfs_closed)
lemma(in UP_cring) lcf_monom:
assumes "a ∈ carrier R"
shows "lcf (monom P a n) = a" "lcf (monom (UP R) a n) = a"
using assms deg_monom cfs_monom apply fastforce
by (metis UP_ring.cfs_monom UP_ring.deg_monom UP_ring_axioms assms)
end
text‹Function which truncates a polynomial by removing the leading term›
definition truncate where
"truncate R f = f ⊖⇘(UP R)⇙ (leading_term R f)"
context UP_ring
begin
abbreviation(input) trunc where
"trunc ≡ truncate R"
lemma trunc_closed:
assumes "f ∈ carrier P"
shows "trunc f ∈ carrier P"
using assms unfolding truncate_def
by (metis ltrm_closed P_def UP_ring.UP_ring UP_ring_axioms leading_term_def ring.ring_simprules(4))
lemma trunc_simps:
assumes "f ∈ carrier P"
shows "f = (trunc f) ⊕⇘P⇙ (ltrm f)"
"f ⊖⇘P⇙ (trunc f) = ltrm f"
apply (metis ltrm_closed P.add.inv_solve_right P.minus_closed P_def a_minus_def assms Cring_Poly.truncate_def leading_term_def)
using trunc_closed[of f] ltrm_closed[of f] P_def P.add.inv_solve_right[of "ltrm f" f "trunc f"]
assms unfolding UP_cring_def
by (metis P.add.inv_closed P.add.m_lcomm P.add.r_inv_ex P.minus_eq P.minus_minus
P.r_neg2 P.r_zero Cring_Poly.truncate_def leading_term_def)
lemma trunc_zero:
assumes "f ∈ carrier P"
assumes "degree f = 0"
shows "trunc f = 𝟬⇘P⇙"
unfolding truncate_def
using assms ltrm_deg_0[of f]
by (metis P.r_neg P_def a_minus_def leading_term_def)
lemma trunc_degree:
assumes "f ∈ carrier P"
assumes "degree f > 0"
shows "degree (trunc f) < degree f"
unfolding truncate_def using assms
by (metis ltrm_closed ltrm_decomp P.add.right_cancel Cring_Poly.truncate_def trunc_closed trunc_simps(1))
text‹The coefficients of trunc agree with f for small degree›
lemma trunc_cfs:
assumes "p ∈ carrier P"
assumes "n < degree p"
shows " (trunc p) n = p n"
using P_def assms(1) assms(2) unfolding truncate_def
by (smt (verit) ltrm_closed ltrm_cfs R.minus_zero R.ring_axioms UP_ring.cfs_minus
UP_ring_axioms a_minus_def cfs_closed leading_term_def nat_neq_iff ring.ring_simprules(15))
text‹monomial predicate›
definition is_UP_monom where
"is_UP_monom = (λf. f ∈ carrier (UP R) ∧ f = ltrm f)"
lemma is_UP_monomI:
assumes "a ∈ carrier R"
assumes "p = monom P a n"
shows "is_UP_monom p"
using assms(1) assms(2) is_UP_monom_def ltrm_monom P_def monom_closed
by auto
lemma is_UP_monomI':
assumes "f ∈ carrier (UP R)"
assumes "f = ltrm f"
shows "is_UP_monom f"
using assms P_def unfolding is_UP_monom_def by blast
lemma monom_is_UP_monom:
assumes "a ∈ carrier R"
shows "is_UP_monom (monom P a n)" "is_UP_monom (monom (UP R) a n)"
using assms P_def ltrm_monom_simp monom_closed
unfolding is_UP_monom_def
by auto
lemma is_UP_monomE:
assumes "is_UP_monom f"
shows "f ∈ carrier P" "f = monom P (lcf f) (degree f)" "f = monom (UP R) (lcf f) (degree f)"
using assms unfolding is_UP_monom_def
by(auto simp: P_def )
lemma ltrm_is_UP_monom:
assumes "p ∈ carrier P"
shows "is_UP_monom (ltrm p)"
using assms
by (simp add: cfs_closed monom_is_UP_monom(1))
lemma is_UP_monom_mult:
assumes "is_UP_monom p"
assumes "is_UP_monom q"
shows "is_UP_monom (p ⊗⇘P⇙ q)"
apply(rule is_UP_monomI')
using assms is_UP_monomE P_def UP_mult_closed
apply simp
using assms is_UP_monomE[of p] is_UP_monomE[of q]
P_def monom_mult
by (metis lcf_closed ltrm_monom R.m_closed)
end
subsection‹Properties of Leading Terms and Leading Coefficients in Commutative Rings and Domains›
context UP_cring
begin
lemma cring_deg_mult:
assumes "q ∈ carrier P"
assumes "p ∈ carrier P"
assumes "lcf q ⊗ lcf p ≠𝟬"
shows "degree (q ⊗⇘P⇙ p) = degree p + degree q"
proof-
have "q ⊗⇘P⇙ p = (trunc q ⊕⇘P⇙ ltrm q) ⊗⇘P⇙ (trunc p ⊕⇘P⇙ ltrm p)"
using assms(1) assms(2) trunc_simps(1) by auto
then have "q ⊗⇘P⇙ p = (trunc q ⊕⇘P⇙ ltrm q) ⊗⇘P⇙ (trunc p ⊕⇘P⇙ ltrm p)"
by linarith
then have 0: "q ⊗⇘P⇙ p = (trunc q ⊗⇘P⇙ (trunc p ⊕⇘P⇙ ltrm p)) ⊕⇘P⇙ ( ltrm q ⊗⇘P⇙ (trunc p ⊕⇘P⇙ ltrm p))"
by (simp add: P.l_distr assms(1) assms(2) ltrm_closed trunc_closed)
have 1: "(trunc q ⊗⇘P⇙ (trunc p ⊕⇘P⇙ ltrm p)) (degree p + degree q) = 𝟬"
proof(cases "degree q = 0")
case True
then show ?thesis
using assms(1) assms(2) trunc_simps(1) trunc_zero by auto
next
case False
have "degree ((trunc q) ⊗⇘P⇙ p) ≤ degree (trunc q) + degree p"
using assms trunc_simps[of q] deg_mult_ring[of "trunc q" p] trunc_closed
by blast
then have "degree (trunc q ⊗⇘P⇙ (trunc p ⊕⇘P⇙ ltrm p)) < degree q + degree p"
using False assms(1) assms(2) trunc_degree trunc_simps(1) by fastforce
then show ?thesis
by (metis P_def UP_mult_closed UP_ring.coeff_simp UP_ring_axioms
add.commute assms(1) assms(2) deg_belowI not_less trunc_closed trunc_simps(1))
qed
have 2: "(q ⊗⇘P⇙ p) (degree p + degree q) =
( ltrm q ⊗⇘P⇙ (trunc p ⊕⇘P⇙ ltrm p)) (degree p + degree q)"
using 0 1 assms cfs_closed trunc_closed by auto
have 3: "(q ⊗⇘P⇙ p) (degree p + degree q) =
( ltrm q ⊗⇘P⇙ trunc p) (degree p + degree q) ⊕ ( ltrm q ⊗⇘P⇙ ltrm p) (degree p + degree q)"
by (simp add: "2" ltrm_closed UP_r_distr assms(1) assms(2) trunc_closed)
have 4: "( ltrm q ⊗⇘P⇙ trunc p) (degree p + degree q) = 𝟬"
proof(cases "degree p = 0")
case True
then show ?thesis
using "2" "3" assms(1) assms(2) cfs_closed ltrm_closed trunc_zero by auto
next
case False
have "degree ( ltrm q ⊗⇘P⇙ trunc p) ≤ degree (ltrm q) + degree (trunc p)"
using assms trunc_simps deg_mult_ring ltrm_closed trunc_closed by presburger
then have "degree ( ltrm q ⊗⇘P⇙ trunc p) < degree q + degree p"
using False assms(1) assms(2) trunc_degree trunc_simps(1) deg_ltrm by fastforce
then show ?thesis
by (metis ltrm_closed P_def UP_mult_closed UP_ring.coeff_simp UP_ring_axioms add.commute assms(1) assms(2) deg_belowI not_less trunc_closed)
qed
have 5: "(q ⊗⇘P⇙ p) (degree p + degree q) = ( ltrm q ⊗⇘P⇙ ltrm p) (degree p + degree q)"
by (simp add: "3" "4" assms(1) assms(2) cfs_closed)
have 6: "ltrm q ⊗⇘P⇙ ltrm p = monom P (lcf q ⊗ lcf p) (degree p + degree q)"
unfolding leading_term_def
by (metis P_def UP_ring.monom_mult UP_ring_axioms add.commute assms(1) assms(2) cfs_closed)
have 7: "( ltrm q ⊗⇘P⇙ ltrm p) (degree p + degree q) ≠𝟬"
using 5 6 assms
by (metis R.m_closed cfs_closed cfs_monom)
have 8: "degree (q ⊗⇘P⇙ p) ≥degree p + degree q"
using 5 6 7 P_def UP_mult_closed assms(1) assms(2)
by (simp add: UP_ring.coeff_simp UP_ring_axioms deg_belowI)
then show ?thesis
using assms(1) assms(2) deg_mult_ring by fastforce
qed
text‹leading term is multiplicative›
lemma ltrm_of_sum_diff_deg:
assumes "q ∈ carrier P"
assumes "a ∈ carrier R"
assumes "a ≠𝟬"
assumes "degree q < n"
assumes "p = q ⊕⇘P⇙ (monom P a n)"
shows "ltrm p = (monom P a n)"
proof-
have 0: "degree (monom P a n) = n"
by (simp add: assms(2) assms(3))
have 1: "(monom P a n) ∈ carrier P"
using assms(2) by auto
have 2: "ltrm ((monom P a n) ⊕⇘P⇙ q) = ltrm (monom P a n)"
using assms ltrm_of_sum_diff_degree[of "(monom P a n)" q] 1 "0" by linarith
then show ?thesis
using UP_a_comm assms(1) assms(2) assms(5) ltrm_monom by auto
qed
lemma(in UP_cring) ltrm_smult_cring:
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
assumes "lcf p ⊗ a ≠ 𝟬"
shows "ltrm (a ⊙⇘P⇙p) = a⊙⇘P⇙(ltrm p)"
using assms
by (smt (verit) lcf_monom(1) P_def R.m_closed R.m_comm cfs_closed cfs_smult coeff_simp
cring_deg_mult deg_monom deg_ltrm monom_closed monom_mult_is_smult monom_mult_smult)
lemma(in UP_cring) deg_zero_ltrm_smult_cring:
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
assumes "degree p = 0"
shows "ltrm (a ⊙⇘P⇙p) = a⊙⇘P⇙(ltrm p)"
by (metis ltrm_deg_0 assms(1) assms(2) assms(3) deg_smult_decr le_0_eq module.smult_closed module_axioms)
lemma(in UP_domain) ltrm_smult:
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
shows "ltrm (a ⊙⇘P⇙p) = a⊙⇘P⇙(ltrm p)"
by (metis lcf_closed ltrm_closed ltrm_smult_cring P_def R.integral_iff UP_ring.deg_ltrm
UP_ring_axioms UP_smult_zero assms(1) assms(2) cfs_zero deg_nzero_nzero deg_zero_ltrm_smult_cring monom_zero)
lemma(in UP_cring) cring_ltrm_mult:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "lcf p ⊗ lcf q ≠ 𝟬"
shows "ltrm (p ⊗⇘P⇙ q) = (ltrm p) ⊗⇘P⇙ (ltrm q)"
proof(cases "degree p = 0 ∨ degree q = 0")
case True
then show ?thesis
by (smt (verit) ltrm_closed ltrm_deg_0 ltrm_smult_cring R.m_comm UP_m_comm assms(1) assms(2) assms(3) cfs_closed monom_mult_is_smult)
next
case False
obtain q0 where q0_def: "q0 = trunc q"
by simp
obtain p0 where p0_def: "p0 = trunc p"
by simp
have Pq: "degree q0 < degree q"
using False P_def assms(2) q0_def trunc_degree by blast
have Pp: "degree p0 < degree p"
using False P_def assms(1) p0_def trunc_degree by blast
have "p ⊗⇘P⇙ q = (p0 ⊕⇘P⇙ ltrm(p)) ⊗⇘P ⇙(q0 ⊕⇘P⇙ ltrm(q))"
using assms(1) assms(2) p0_def q0_def trunc_simps(1) by auto
then have P0: "p ⊗⇘P⇙ q = ((p0 ⊕⇘P⇙ ltrm(p)) ⊗⇘P ⇙q0) ⊕⇘P⇙ ((p0 ⊕⇘P⇙ ltrm(p))⊗⇘P ⇙ltrm(q))"
by (simp add: P.r_distr assms(1) assms(2) ltrm_closed p0_def q0_def trunc_closed)
have P1: "degree ((p0 ⊕⇘P⇙ ltrm(p)) ⊗⇘P ⇙q0) < degree ((p0 ⊕⇘P⇙ ltrm(p))⊗⇘P ⇙ltrm(q))"
proof-
have LHS: "degree ((p0 ⊕⇘P⇙ ltrm(p)) ⊗⇘P ⇙q0) ≤ degree p + degree q0 "
proof(cases "q0 = 𝟬⇘P⇙")
case True
then show ?thesis
using assms(1) p0_def trunc_simps(1) by auto
next
case False
then show ?thesis
using assms(1) assms(2) deg_mult_ring p0_def
q0_def trunc_simps(1) trunc_closed by auto
qed
have RHS: "degree ((p0 ⊕⇘P⇙ ltrm(p))⊗⇘P ⇙ltrm(q)) = degree p + degree q"
using assms(1) assms(2) deg_mult_ring ltrm_closed p0_def trunc_simps(1)
by (smt (verit) P_def UP_cring.lcf_monom(1) UP_cring.cring_deg_mult UP_cring_axioms add.commute assms(3) cfs_closed deg_ltrm)
then show ?thesis
using RHS LHS Pq
by linarith
qed
then have P2: "ltrm (p ⊗⇘P⇙ q) = ltrm ((p0 ⊕⇘P⇙ ltrm(p))⊗⇘P ⇙ltrm(q))"
using P0 P1
by (metis (no_types, lifting) ltrm_closed ltrm_of_sum_diff_degree P.add.m_comm
UP_mult_closed assms(1) assms(2) p0_def q0_def trunc_closed trunc_simps(1))
have P3: " ltrm ((p0 ⊕⇘P⇙ ltrm(p))⊗⇘P ⇙ltrm(q)) = ltrm p ⊗⇘P⇙ ltrm q"
proof-
have Q0: "((p0 ⊕⇘P⇙ ltrm(p))⊗⇘P ⇙ltrm(q)) = (p0 ⊗⇘P ⇙ltrm(q)) ⊕⇘P⇙ (ltrm(p))⊗⇘P ⇙ltrm(q)"
by (simp add: P.l_distr assms(1) assms(2) ltrm_closed p0_def trunc_closed)
have Q1: "degree ((p0 ⊗⇘P ⇙ltrm(q)) ) < degree ((ltrm(p))⊗⇘P ⇙ltrm(q))"
proof(cases "p0 = 𝟬⇘P⇙")
case True
then show ?thesis
using P1 assms(1) assms(2) ltrm_closed by auto
next
case F: False
then show ?thesis
proof-
have LHS: "degree ((p0 ⊗⇘P ⇙ltrm(q))) < degree p + degree q"
using False F Pp assms(1) assms(2) deg_nzero_nzero
deg_ltrm ltrm_closed p0_def trunc_closed
by (smt (verit) add_le_cancel_right deg_mult_ring le_trans not_less)
have RHS: "degree ((ltrm(p))⊗⇘P ⇙ltrm(q)) = degree p + degree q"
using cring_deg_mult[of "ltrm p" "ltrm q"] assms
by (simp add: ltrm_closed ltrm_cfs deg_ltrm)
then show ?thesis using LHS RHS by auto
qed
qed
have Q2: "ltrm ((p0 ⊕⇘P⇙ ltrm(p))⊗⇘P ⇙ltrm(q)) = ltrm ((ltrm(p))⊗⇘P ⇙ltrm(q))"
using Q0 Q1
by (metis (no_types, lifting) ltrm_closed ltrm_of_sum_diff_degree P.add.m_comm
UP_mult_closed assms(1) assms(2) p0_def trunc_closed)
show ?thesis using ltrm_prod_ltrm Q0 Q1 Q2
by (simp add: assms(1) assms(2))
qed
then show ?thesis
by (simp add: P2)
qed
lemma(in UP_domain) ltrm_mult:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
shows "ltrm (p ⊗⇘P⇙ q) = (ltrm p) ⊗⇘P⇙ (ltrm q)"
using cring_ltrm_mult assms
by (smt (verit) ltrm_closed ltrm_deg_0 cfs_closed deg_nzero_nzero deg_ltrm local.integral_iff monom_mult monom_zero)
lemma lcf_deg_0:
assumes "degree p = 0"
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
shows "(p ⊗⇘P⇙ q) = (lcf p)⊙⇘P⇙q"
using P_def assms(1) assms(2) assms(3)
by (metis ltrm_deg_0 cfs_closed monom_mult_is_smult)
text‹leading term powers›
lemma (in domain) nonzero_pow_nonzero:
assumes "a ∈ carrier R"
assumes "a ≠𝟬"
shows "a[^](n::nat) ≠ 𝟬"
proof(induction n)
case 0
then show ?case
by auto
next
case (Suc n)
fix n::nat
assume IH: "a[^] n ≠ 𝟬"
show "a[^] (Suc n) ≠ 𝟬"
proof-
have "a[^] (Suc n) = a[^] n ⊗ a"
by simp
then show ?thesis using assms IH
using IH assms(1) assms(2) local.integral by auto
qed
qed
lemma (in UP_cring) cring_monom_degree:
assumes "a ∈ (carrier R)"
assumes "p = monom P a m"
assumes "a[^]n ≠ 𝟬"
shows "degree (p[^]⇘P⇙ n) = n*m"
by (simp add: assms(1) assms(2) assms(3) monom_pow)
lemma (in UP_domain) monom_degree:
assumes "a ≠𝟬"
assumes "a ∈ (carrier R)"
assumes "p = monom P a m"
shows "degree (p[^]⇘P⇙ n) = n*m"
by (simp add: R.domain_axioms assms(1) assms(2) assms(3) domain.nonzero_pow_nonzero monom_pow)
lemma(in UP_cring) cring_pow_ltrm:
assumes "p ∈ carrier P"
assumes "lcf p [^]n ≠ 𝟬"
shows "ltrm (p[^]⇘P⇙(n::nat)) = (ltrm p)[^]⇘P⇙n"
proof-
have "lcf p [^]n ≠ 𝟬 ⟹ ltrm (p[^]⇘P⇙(n::nat)) = (ltrm p)[^]⇘P⇙n"
proof(induction n)
case 0
then show ?case
using P.ring_simprules(6) P.nat_pow_0 cfs_one deg_one monom_one by presburger
next
case (Suc n) fix n::nat
assume IH : "(lcf p [^] n ≠ 𝟬 ⟹ ltrm (p [^]⇘P⇙ n) = ltrm p [^]⇘P⇙ n)"
assume A: "lcf p [^] Suc n ≠ 𝟬"
have a: "ltrm (p [^]⇘P⇙ n) = ltrm p [^]⇘P⇙ n"
apply(cases "lcf p [^] n = 𝟬")
using A lcf_closed assms(1) apply auto[1]
by(rule IH)
have 0: "lcf (ltrm (p [^]⇘P⇙ n)) = lcf p [^] n"
unfolding a
by (simp add: lcf_monom(1) assms(1) cfs_closed monom_pow)
then have 1: "lcf (ltrm (p [^]⇘P⇙ n)) ⊗ lcf p ≠ 𝟬"
using assms A R.nat_pow_Suc IH by metis
then show "ltrm (p [^]⇘P⇙ Suc n) = ltrm p [^]⇘P⇙ Suc n"
using IH 0 assms(1) cring_ltrm_mult cfs_closed
by (smt (verit) A lcf_monom(1) ltrm_closed P.nat_pow_Suc2 P.nat_pow_closed R.nat_pow_Suc2 a)
qed
then show ?thesis
using assms(2) by blast
qed
lemma(in UP_cring) cring_pow_deg:
assumes "p ∈ carrier P"
assumes "lcf p [^]n ≠ 𝟬"
shows "degree (p[^]⇘P⇙(n::nat)) = n*degree p"
proof-
have "degree ( (ltrm p)[^]⇘P⇙n) = n*degree p"
using assms(1) assms(2) cring_monom_degree lcf_closed lcf_ltrm by auto
then show ?thesis
using assms cring_pow_ltrm
by (metis P.nat_pow_closed P_def UP_ring.deg_ltrm UP_ring_axioms)
qed
lemma(in UP_cring) cring_pow_deg_bound:
assumes "p ∈ carrier P"
shows "degree (p[^]⇘P⇙(n::nat)) ≤ n*degree p"
apply(induction n)
apply (metis Group.nat_pow_0 deg_one le_zero_eq mult_is_0)
using deg_mult_ring[of _ p]
by (smt (verit) P.nat_pow_Suc2 P.nat_pow_closed ab_semigroup_add_class.add_ac(1) assms deg_mult_ring le_iff_add mult_Suc)
lemma(in UP_cring) deg_smult:
assumes "a ∈ carrier R"
assumes "f ∈ carrier (UP R)"
assumes "a ⊗ lcf f ≠ 𝟬"
shows "deg R (a ⊙⇘UP R⇙ f) = deg R f"
using assms P_def cfs_smult deg_eqI deg_smult_decr smult_closed
by (metis deg_gtE le_neq_implies_less)
lemma(in UP_cring) deg_smult':
assumes "a ∈ Units R"
assumes "f ∈ carrier (UP R)"
shows "deg R (a ⊙⇘UP R⇙ f) = deg R f"
apply(cases "deg R f = 0")
apply (metis P_def R.Units_closed assms(1) assms(2) deg_smult_decr le_zero_eq)
apply(rule deg_smult)
using assms apply blast
using assms apply blast
proof
assume A: "deg R f ≠ 0" "a ⊗ f (deg R f) = 𝟬"
have 0: "f (deg R f) = 𝟬"
using A assms R.Units_not_right_zero_divisor[of a "f (deg R f)"] UP_car_memE(1) by blast
then show False using assms A
by (metis P_def deg_zero deg_ltrm monom_zero)
qed
lemma(in UP_domain) pow_sum0:
"⋀ p q. p ∈ carrier P ⟹ q ∈ carrier P ⟹ degree q < degree p ⟹ degree ((p ⊕⇘P⇙ q )[^]⇘P⇙n) = (degree p)*n"
proof(induction n)
case 0
then show ?case
by (metis Group.nat_pow_0 deg_one mult_is_0)
next
case (Suc n)
fix n
assume IH: "⋀ p q. p ∈ carrier P ⟹ q ∈ carrier P ⟹
degree q < degree p ⟹ degree ((p ⊕⇘P⇙ q )[^]⇘P⇙n) = (degree p)*n"
then show "⋀ p q. p ∈ carrier P ⟹ q ∈ carrier P ⟹
degree q < degree p ⟹ degree ((p ⊕⇘P⇙ q )[^]⇘P⇙(Suc n)) = (degree p)*(Suc n)"
proof-
fix p q
assume A0: "p ∈ carrier P" and
A1: "q ∈ carrier P" and
A2: "degree q < degree p"
show "degree ((p ⊕⇘P⇙ q )[^]⇘P⇙(Suc n)) = (degree p)*(Suc n)"
proof(cases "q = 𝟬⇘P⇙")
case True
then show ?thesis
by (metis A0 A1 A2 IH P.nat_pow_Suc2 P.nat_pow_closed P.r_zero deg_mult
domain.nonzero_pow_nonzero local.domain_axioms mult_Suc_right nat_neq_iff)
next
case False
then show ?thesis
proof-
have P0: "degree ((p ⊕⇘P⇙ q )[^]⇘P⇙n) = (degree p)*n"
using A0 A1 A2 IH by auto
have P1: "(p ⊕⇘P⇙ q )[^]⇘P⇙(Suc n) = ((p ⊕⇘P⇙ q )[^]⇘P⇙n) ⊗⇘P⇙ (p ⊕⇘P⇙ q )"
by simp
then have P2: "(p ⊕⇘P⇙ q )[^]⇘P⇙(Suc n) = (((p ⊕⇘P⇙ q )[^]⇘P⇙n) ⊗⇘P⇙ p) ⊕⇘P⇙ (((p ⊕⇘P⇙ q )[^]⇘P⇙n) ⊗⇘P⇙ q)"
by (simp add: A0 A1 UP_r_distr)
have P3: "degree (((p ⊕⇘P⇙ q )[^]⇘P⇙n) ⊗⇘P⇙ p) = (degree p)*n + (degree p)"
using P0 A0 A1 A2 deg_nzero_nzero degree_of_sum_diff_degree local.nonzero_pow_nonzero by auto
have P4: "degree (((p ⊕⇘P⇙ q )[^]⇘P⇙n) ⊗⇘P⇙ q) = (degree p)*n + (degree q)"
using P0 A0 A1 A2 deg_nzero_nzero degree_of_sum_diff_degree local.nonzero_pow_nonzero False deg_mult
by simp
have P5: "degree (((p ⊕⇘P⇙ q )[^]⇘P⇙n) ⊗⇘P⇙ p) > degree (((p ⊕⇘P⇙ q )[^]⇘P⇙n) ⊗⇘P⇙ q)"
using P3 P4 A2 by auto
then show ?thesis using P5 P3 P2
by (simp add: A0 A1 degree_of_sum_diff_degree)
qed
qed
qed
qed
lemma(in UP_domain) pow_sum:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "degree q < degree p"
shows "degree ((p ⊕⇘P⇙ q )[^]⇘P⇙n) = (degree p)*n"
using assms(1) assms(2) assms(3) pow_sum0 by blast
lemma(in UP_domain) deg_pow0:
"⋀ p. p ∈ carrier P ⟹ n ≥ degree p ⟹ degree (p [^]⇘P⇙ m) = m*(degree p)"
proof(induction n)
case 0
show "p ∈ carrier P ⟹ 0 ≥ degree p ⟹ degree (p [^]⇘P⇙ m) = m*(degree p)"
proof-
assume B0:"p ∈ carrier P"
assume B1: "0 ≥ degree p"
then obtain a where a_def: "a ∈ carrier R ∧ p = monom P a 0"
using B0 deg_zero_impl_monom by fastforce
show "degree (p [^]⇘P⇙ m) = m*(degree p)" using UP_cring.monom_pow
by (metis P_def R.nat_pow_closed UP_cring_axioms a_def deg_const
mult_0_right mult_zero_left)
qed
next
case (Suc n)
fix n
assume IH: "⋀p. (p ∈ carrier P ⟹ n ≥degree p ⟹ degree (p [^]⇘P⇙ m) = m * (degree p))"
show "p ∈ carrier P ⟹ Suc n ≥ degree p ⟹ degree (p [^]⇘P⇙ m) = m * (degree p)"
proof-
assume A0: "p ∈ carrier P"
assume A1: "Suc n ≥ degree p"
show "degree (p [^]⇘P⇙ m) = m * (degree p)"
proof(cases "Suc n > degree p")
case True
then show ?thesis using IH A0 by simp
next
case False
then show ?thesis
proof-
obtain q where q_def: "q = trunc p"
by simp
obtain k where k_def: "k = degree q"
by simp
have q_is_poly: "q ∈ carrier P"
by (simp add: A0 q_def trunc_closed)
have k_bound0: "k <degree p"
using k_def q_def trunc_degree[of p] A0 False by auto
have k_bound1: "k ≤ n"
using k_bound0 A0 A1 by auto
have P_q:"degree (q [^]⇘P⇙ m) = m * k"
using IH[of "q"] k_bound1 k_def q_is_poly by auto
have P_ltrm: "degree ((ltrm p) [^]⇘P⇙ m) = m*(degree p)"
proof-
have "degree p = degree (ltrm p)"
by (simp add: A0 deg_ltrm)
then show ?thesis using monom_degree
by (metis A0 P.r_zero P_def cfs_closed coeff_simp equal_deg_sum k_bound0 k_def lcoeff_nonzero2 nat_neq_iff q_is_poly)
qed
have "p = q ⊕⇘P⇙ (ltrm p)"
by (simp add: A0 q_def trunc_simps(1))
then show ?thesis
using P_q pow_sum[of "(ltrm p)" q m] A0 UP_a_comm
deg_ltrm k_bound0 k_def ltrm_closed q_is_poly by auto
qed
qed
qed
qed
lemma(in UP_domain) deg_pow:
assumes "p ∈ carrier P"
shows "degree (p [^]⇘P⇙ m) = m*(degree p)"
using deg_pow0 assms by blast
lemma(in UP_domain) ltrm_pow0:
"⋀f. f ∈ carrier P ⟹ ltrm (f [^]⇘P⇙ (n::nat)) = (ltrm f) [^]⇘P⇙ n"
proof(induction n)
case 0
then show ?case
using ltrm_deg_0 P.nat_pow_0 P.ring_simprules(6) deg_one by presburger
next
case (Suc n)
fix n::nat
assume IH: "⋀f. f ∈ carrier P ⟹ ltrm (f [^]⇘P⇙ n) = (ltrm f) [^]⇘P⇙ n"
then show "⋀f. f ∈ carrier P ⟹ ltrm (f [^]⇘P⇙ (Suc n)) = (ltrm f) [^]⇘P⇙ (Suc n)"
proof-
fix f
assume A: "f ∈ carrier P"
show " ltrm (f [^]⇘P⇙ (Suc n)) = (ltrm f) [^]⇘P⇙ (Suc n)"
proof-
have 0: "ltrm (f [^]⇘P⇙ n) = (ltrm f) [^]⇘P⇙ n"
using A IH by blast
have 1: "ltrm (f [^]⇘P⇙ (Suc n)) = ltrm ((f [^]⇘P⇙ n)⊗⇘P⇙ f)"
by auto then
show ?thesis using ltrm_mult 0 1
by (simp add: A)
qed
qed
qed
lemma(in UP_domain) ltrm_pow:
assumes "f ∈ carrier P"
shows " ltrm (f [^]⇘P⇙ (n::nat)) = (ltrm f) [^]⇘P⇙ n"
using assms ltrm_pow0 by blast
text‹lemma on the leading coefficient›
lemma lcf_eq:
assumes "f ∈ carrier P"
shows "lcf f = lcf (ltrm f)"
using ltrm_deg_0
by (simp add: ltrm_cfs assms deg_ltrm)
lemma lcf_eq_deg_eq_imp_ltrm_eq:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "degree p > 0"
assumes "degree p = degree q"
assumes "lcf p = lcf q"
shows "ltrm p = ltrm q"
using assms(4) assms(5)
by (simp add: leading_term_def)
lemma ltrm_eq_imp_lcf_eq:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "ltrm p = ltrm q"
shows "lcf p = lcf q"
using assms
by (metis lcf_eq)
lemma ltrm_eq_imp_deg_drop:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "ltrm p = ltrm q"
assumes "degree p >0"
shows "degree (p ⊖⇘P⇙ q) < degree p"
proof-
have P0: "degree p = degree q"
by (metis assms(1) assms(2) assms(3) deg_ltrm)
then have P1: "degree (p ⊖⇘P⇙ q) ≤ degree p"
by (metis P.add.inv_solve_right P.minus_closed P.minus_eq assms(1)
assms(2) degree_of_sum_diff_degree neqE order.strict_implies_order order_refl)
have "degree (p ⊖⇘P⇙ q) ≠ degree p"
proof
assume A: "degree (p ⊖⇘P⇙ q) = degree p"
have Q0: "p ⊖⇘P⇙ q = ((trunc p) ⊕⇘P⇙ (ltrm p)) ⊖⇘P⇙ ((trunc q) ⊕⇘P⇙ (ltrm p))"
using assms(1) assms(2) assms(3) trunc_simps(1) by force
have Q1: "p ⊖⇘P⇙ q = (trunc p) ⊖⇘P⇙ (trunc q)"
proof-
have "p ⊖⇘P⇙ q = ((trunc p) ⊕⇘P⇙ (ltrm p)) ⊖⇘P⇙ (trunc q) ⊖ ⇘P⇙ (ltrm p)"
using Q0
by (simp add: P.minus_add P.minus_eq UP_a_assoc assms(1) assms(2) ltrm_closed trunc_closed)
then show ?thesis
by (metis (no_types, lifting) ltrm_closed P.add.inv_mult_group P.minus_eq
P.r_neg2 UP_a_assoc assms(1) assms(2) assms(3) carrier_is_submodule submoduleE(6) trunc_closed trunc_simps(1))
qed
have Q2: "degree (trunc p) < degree p"
by (simp add: assms(1) assms(4) trunc_degree)
have Q3: "degree (trunc q) < degree q"
using P0 assms(2) assms(4) trunc_degree by auto
then show False using A Q1 Q2 Q3 by (simp add: P.add.inv_solve_right
P.minus_eq P0 assms(1) assms(2) degree_of_sum_diff_degree trunc_closed)
qed
then show ?thesis
using P1 by auto
qed
lemma(in UP_cring) cring_lcf_scalar_mult:
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
assumes "a ⊗ (lcf p) ≠𝟬"
shows "lcf (a ⊙⇘P⇙ p) = a ⊗ (lcf p)"
proof-
have 0: "lcf (a ⊙⇘P⇙ p) = lcf (ltrm (a ⊙⇘P⇙ p))"
using assms lcf_eq smult_closed by blast
have 1: "degree (a ⊙⇘P⇙ p) = degree p"
by (smt (verit) lcf_monom(1) P_def R.one_closed R.r_null UP_ring.coeff_smult UP_ring_axioms
assms(1) assms(2) assms(3) coeff_simp cring_deg_mult deg_const monom_closed monom_mult_is_smult smult_one)
then have "lcf (a ⊙⇘P⇙ p) = lcf (a ⊙⇘P⇙ (ltrm p))"
using lcf_eq[of "a ⊙⇘P⇙ p"] smult_closed assms 0
by (metis cfs_closed cfs_smult monom_mult_smult)
then show ?thesis
unfolding leading_term_def
by (metis P_def R.m_closed UP_cring.lcf_monom UP_cring_axioms assms(1) assms(2) cfs_closed monom_mult_smult)
qed
lemma(in UP_domain) lcf_scalar_mult:
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
shows "lcf (a ⊙⇘P⇙ p) = a ⊗ (lcf p)"
proof-
have "lcf (a ⊙⇘P⇙ p) = lcf (ltrm (a ⊙⇘P⇙ p))"
using lcf_eq UP_smult_closed assms(1) assms(2) by blast
then have "lcf (a ⊙⇘P⇙ p) = lcf (a ⊙⇘P⇙ (ltrm p))"
using ltrm_smult assms(1) assms(2) by metis
then show ?thesis
by (metis (full_types) UP_smult_zero assms(1) assms(2) cfs_smult cfs_zero deg_smult)
qed
lemma(in UP_cring) cring_lcf_mult:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "(lcf p) ⊗ (lcf q) ≠𝟬"
shows "lcf (p ⊗⇘P⇙ q) = (lcf p) ⊗ (lcf q)"
using assms cring_ltrm_mult
by (smt (verit) lcf_monom(1) P.m_closed R.m_closed cfs_closed monom_mult)
lemma(in UP_domain) lcf_mult:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
shows "lcf (p ⊗⇘P⇙ q) = (lcf p) ⊗ (lcf q)"
by (metis P_def R.integral_iff assms(1) assms(2) cfs_closed coeff_simp cring_lcf_mult lcoeff_nonzero local.integral_iff)
lemma(in UP_cring) cring_lcf_pow:
assumes "p ∈ carrier P"
assumes "(lcf p)[^]n ≠𝟬"
shows "lcf (p[^]⇘P⇙(n::nat)) = (lcf p)[^]n"
by (smt (verit) P.nat_pow_closed R.nat_pow_closed assms(1) assms(2) cring_pow_ltrm lcf_closed lcf_ltrm lcf_monom monom_pow)
lemma(in UP_domain) lcf_pow:
assumes "p ∈ carrier P"
shows "lcf (p[^]⇘P⇙(n::nat)) = (lcf p)[^]n"
proof-
show ?thesis
proof(induction n)
case 0
then show ?case
by (metis Group.nat_pow_0 P_def R.one_closed UP_cring.lcf_monom UP_cring_axioms monom_one)
next
case (Suc n)
fix n
assume IH: "lcf (p[^]⇘P⇙(n::nat)) = (lcf p)[^]n"
show "lcf (p[^]⇘P⇙(Suc n)) = (lcf p)[^](Suc n)"
proof-
have "lcf (p[^]⇘P⇙(Suc n)) = lcf ((p[^]⇘P⇙n) ⊗⇘P⇙p)"
by simp
then have "lcf (p[^]⇘P⇙(Suc n)) = (lcf p)[^]n ⊗ (lcf p)"
by (simp add: IH assms lcf_mult)
then show ?thesis by auto
qed
qed
qed
end
subsection‹Constant Terms and Constant Coefficients›
text‹Constant term and coefficient function›
definition zcf where
"zcf f = (f 0)"
abbreviation(in UP_cring)(input) ctrm where
"ctrm f ≡ monom P (f 0) 0"
context UP_cring
begin
lemma ctrm_is_poly:
assumes "p ∈ carrier P"
shows "ctrm p ∈ carrier P"
by (simp add: assms cfs_closed)
lemma ctrm_degree:
assumes "p ∈ carrier P"
shows "degree (ctrm p) = 0"
by (simp add: assms cfs_closed)
lemma ctrm_zcf:
assumes "f ∈ carrier P"
assumes "zcf f = 𝟬"
shows "ctrm f = 𝟬⇘P⇙"
by (metis P_def UP_ring.monom_zero UP_ring_axioms zcf_def assms(2))
lemma zcf_degree_zero:
assumes "f ∈ carrier P"
assumes "degree f = 0"
shows "lcf f = zcf f"
by (simp add: zcf_def assms(2))
lemma zcf_zero_degree_zero:
assumes "f ∈ carrier P"
assumes "degree f = 0"
assumes "zcf f = 𝟬"
shows "f = 𝟬⇘P⇙"
using zcf_degree_zero[of f] assms ltrm_deg_0[of f]
by simp
lemma zcf_ctrm:
assumes "p ∈ carrier P"
shows "zcf (ctrm p) = zcf p"
unfolding zcf_def
using P_def UP_ring.cfs_monom UP_ring_axioms assms cfs_closed by fastforce
lemma ctrm_trunc:
assumes "p ∈ carrier P"
assumes "degree p >0"
shows "zcf(trunc p) = zcf p"
by (simp add: zcf_def assms(1) assms(2) trunc_cfs)
text‹Constant coefficient function is a ring homomorphism›
lemma zcf_add:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
shows "zcf(p ⊕⇘P⇙ q) = (zcf p) ⊕ (zcf q)"
by (simp add: zcf_def assms(1) assms(2))
lemma coeff_ltrm[simp]:
assumes "p ∈ carrier P"
assumes "degree p > 0"
shows "zcf(ltrm p) = 𝟬"
by (metis ltrm_cfs_above_deg ltrm_cfs zcf_def assms(1) assms(2))
lemma zcf_zero[simp]:
"zcf 𝟬⇘P⇙ = 𝟬"
using zcf_degree_zero by auto
lemma zcf_one[simp]:
"zcf 𝟭⇘P⇙ = 𝟭"
by (simp add: zcf_def)
lemma ctrm_smult:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "ctrm (a ⊙⇘P⇙ f) = a ⊙⇘P⇙(ctrm f)"
using P_def UP_ring.monom_mult_smult UP_ring_axioms assms(1) assms(2) cfs_smult coeff_simp
by (simp add: UP_ring.monom_mult_smult cfs_closed)
lemma ctrm_monom[simp]:
assumes "a ∈ carrier R"
shows "ctrm (monom P a (Suc k)) = 𝟬⇘P⇙"
by (simp add: assms cfs_monom)
end
subsection‹Polynomial Induction Rules›
context UP_ring
begin
text‹Rule for strong induction on polynomial degree›
lemma poly_induct:
assumes "p ∈ carrier P"
assumes Deg_0: "⋀p. p ∈ carrier P ⟹ degree p = 0 ⟹ Q p"
assumes IH: "⋀p. (⋀q. q ∈ carrier P ⟹ degree q < degree p ⟹ Q q) ⟹ p ∈ carrier P ⟹ degree p > 0 ⟹ Q p"
shows "Q p"
proof-
have "⋀n. ⋀p. p ∈ carrier P ⟹ degree p ≤ n ⟹ Q p"
proof-
fix n
show "⋀p. p ∈ carrier P ⟹ degree p ≤ n ⟹ Q p"
proof(induction n)
case 0
then show ?case
using Deg_0 by simp
next
case (Suc n)
fix n
assume I: "⋀p. p ∈ carrier P ⟹ degree p ≤ n ⟹ Q p"
show "⋀p. p ∈ carrier P ⟹ degree p ≤ (Suc n) ⟹ Q p"
proof-
fix p
assume A0: " p ∈ carrier P "
assume A1: "degree p ≤Suc n"
show "Q p"
proof(cases "degree p < Suc n")
case True
then show ?thesis
using I A0 by auto
next
case False
then have D: "degree p = Suc n"
by (simp add: A1 nat_less_le)
then have "(⋀q. q ∈ carrier P ⟹ degree q < degree p ⟹ Q q)"
using I by simp
then show "Q p"
using IH D A0 A1 Deg_0 by blast
qed
qed
qed
qed
then show ?thesis using assms by blast
qed
text‹Variant on induction on degree›
lemma poly_induct2:
assumes "p ∈ carrier P"
assumes Deg_0: "⋀p. p ∈ carrier P ⟹ degree p = 0 ⟹ Q p"
assumes IH: "⋀p. degree p > 0 ⟹ p ∈ carrier P ⟹ Q (trunc p) ⟹ Q p"
shows "Q p"
proof(rule poly_induct)
show "p ∈ carrier P"
by (simp add: assms(1))
show "⋀p. p ∈ carrier P ⟹ degree p = 0 ⟹ Q p"
by (simp add: Deg_0)
show "⋀p. (⋀q. q ∈ carrier P ⟹ degree q < degree p ⟹ Q q) ⟹ p ∈ carrier P ⟹ 0 < degree p ⟹ Q p"
proof-
fix p
assume A0: "(⋀q. q ∈ carrier P ⟹ degree q < degree p ⟹ Q q)"
assume A1: " p ∈ carrier P"
assume A2: "0 < degree p"
show "Q p"
proof-
have "degree (trunc p) < degree p"
by (simp add: A1 A2 trunc_degree)
have "Q (trunc p)"
by (simp add: A0 A1 ‹degree (trunc p) < degree p› trunc_closed)
then show ?thesis
by (simp add: A1 A2 IH)
qed
qed
qed
text‹Additive properties which are true for all monomials are true for all polynomials ›
lemma poly_induct3:
assumes "p ∈ carrier P"
assumes add: "⋀p q. q ∈ carrier P ⟹ p ∈ carrier P ⟹ Q p ⟹ Q q ⟹ Q (p ⊕⇘P⇙ q)"
assumes monom: "⋀a n. a ∈ carrier R ⟹ Q (monom P a n)"
shows "Q p"
apply(rule poly_induct2)
apply (simp add: assms(1))
apply (metis lcf_closed P_def coeff_simp deg_zero_impl_monom monom)
by (metis lcf_closed ltrm_closed add monom trunc_closed trunc_simps(1))
lemma poly_induct4:
assumes "p ∈ carrier P"
assumes add: "⋀p q. q ∈ carrier P ⟹ p ∈ carrier P ⟹ Q p ⟹ Q q ⟹ Q (p ⊕⇘P⇙ q)"
assumes monom_zero: "⋀a. a ∈ carrier R ⟹ Q (monom P a 0)"
assumes monom_Suc: "⋀a n. a ∈ carrier R ⟹ Q (monom P a (Suc n))"
shows "Q p"
apply(rule poly_induct3)
using assms(1) apply auto[1]
using add apply blast
using monom_zero monom_Suc
by (metis P_def UP_ring.monom_zero UP_ring_axioms deg_monom deg_monom_le le_0_eq le_SucE zero_induct)
lemma monic_monom_smult:
assumes "a ∈ carrier R"
shows "a ⊙⇘P⇙ monom P 𝟭 n = monom P a n"
using assms
by (metis R.one_closed R.r_one monom_mult_smult)
lemma poly_induct5:
assumes "p ∈ carrier P"
assumes add: "⋀p q. q ∈ carrier P ⟹ p ∈ carrier P ⟹ Q p ⟹ Q q ⟹ Q (p ⊕⇘P⇙ q)"
assumes monic_monom: "⋀n. Q (monom P 𝟭 n)"
assumes smult: "⋀p a . a ∈ carrier R ⟹ p ∈ carrier P ⟹ Q p ⟹ Q (a ⊙⇘P⇙ p)"
shows "Q p"
apply(rule poly_induct3)
apply (simp add: assms(1))
using add apply blast
proof-
fix a n assume A: "a ∈ carrier R" show "Q (monom P a n)"
using monic_monom[of n] smult[of a "monom P 𝟭 n"] monom_mult_smult[of a 𝟭 n]
by (simp add: A)
qed
lemma poly_induct6:
assumes "p ∈ carrier P"
assumes monom: "⋀a n. a ∈ carrier R ⟹ Q (monom P a 0)"
assumes plus_monom: "⋀a n p. a ∈ carrier R ⟹ a ≠ 𝟬 ⟹ p ∈ carrier P ⟹ degree p < n ⟹ Q p ⟹
Q(p ⊕⇘P⇙ monom P a n)"
shows "Q p"
apply(rule poly_induct2)
using assms(1) apply auto[1]
apply (metis lcf_closed P_def coeff_simp deg_zero_impl_monom monom)
using plus_monom
by (metis lcf_closed P_def coeff_simp lcoeff_nonzero_deg nat_less_le trunc_closed trunc_degree trunc_simps(1))
end
section‹Mapping a Polynomial to its Associated Ring Function›
text‹Turning a polynomial into a function on R:›
definition to_function where
"to_function S f = (λs ∈ carrier S. eval S S (λx. x) s f)"
context UP_cring
begin
definition to_fun where
"to_fun f ≡ to_function R f"
text‹Explicit formula for evaluating a polynomial function:›
lemma to_fun_eval:
assumes "f ∈ carrier P"
assumes "x ∈ carrier R"
shows "to_fun f x = eval R R (λx. x) x f"
using assms unfolding to_function_def to_fun_def
by auto
lemma to_fun_formula:
assumes "f ∈ carrier P"
assumes "x ∈ carrier R"
shows "to_fun f x = (⨁i ∈ {..degree f}. (f i) ⊗ x [^] i)"
proof-
have "f ∈ carrier (UP R)"
using assms P_def by auto
then have "eval R R (λx. x) x f = (⨁⇘R⇙i∈{..deg R f}. (λx. x) (coeff (UP R) f i) ⊗⇘R⇙ x [^]⇘R⇙ i)"
apply(simp add:UnivPoly.eval_def) done
then have "to_fun f x = (⨁⇘R⇙i∈{..deg R f}. (λx. x) (coeff (UP R) f i) ⊗⇘R⇙ x [^]⇘R⇙ i)"
using to_function_def assms unfolding to_fun_def
by (simp add: to_function_def)
then show ?thesis
by(simp add: assms coeff_simp)
qed
lemma eval_ring_hom:
assumes "a ∈ carrier R"
shows "eval R R (λx. x) a ∈ ring_hom P R"
proof-
have "(λx. x) ∈ ring_hom R R"
apply(rule ring_hom_memI)
apply auto done
then have "UP_pre_univ_prop R R (λx. x)"
using R_cring UP_pre_univ_propI by blast
then show ?thesis
by (simp add: P_def UP_pre_univ_prop.eval_ring_hom assms)
qed
lemma to_fun_closed:
assumes "f ∈ carrier P"
assumes "x ∈ carrier R"
shows "to_fun f x ∈ carrier R"
using assms to_fun_eval[of f x] eval_ring_hom[of x]
ring_hom_closed
by fastforce
lemma to_fun_plus:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "x ∈ carrier R"
shows "to_fun (f ⊕⇘P⇙ g) x = (to_fun f x) ⊕ (to_fun g x)"
using assms to_fun_eval[of ] eval_ring_hom[of x]
by (simp add: ring_hom_add)
lemma to_fun_mult:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "x ∈ carrier R"
shows "to_fun (f ⊗⇘P⇙ g) x = (to_fun f x) ⊗ (to_fun g x)"
using assms to_fun_eval[of ] eval_ring_hom[of x]
by (simp add: ring_hom_mult)
lemma to_fun_ring_hom:
assumes "a ∈ carrier R"
shows "(λp. to_fun p a) ∈ ring_hom P R"
apply(rule ring_hom_memI)
apply (simp add: assms to_fun_closed)
apply (simp add: assms to_fun_mult)
apply (simp add: assms to_fun_plus)
using to_fun_eval[of "𝟭⇘P⇙" a] eval_ring_hom[of a]
ring_hom_closed
by (simp add: assms ring_hom_one)
lemma ring_hom_uminus:
assumes "ring S"
assumes "f ∈ (ring_hom S R)"
assumes "a ∈ carrier S"
shows "f (⊖⇘S⇙ a) = ⊖ (f a)"
proof-
have "f (a ⊖⇘S⇙ a) = (f a) ⊕ f (⊖⇘S⇙ a)"
unfolding a_minus_def
by (simp add: assms(1) assms(2) assms(3) ring.ring_simprules(3) ring_hom_add)
then have "(f a) ⊕ f (⊖⇘S⇙ a) = 𝟬 "
by (metis R.ring_axioms a_minus_def assms(1) assms(2) assms(3)
ring.ring_simprules(16) ring_hom_zero)
then show ?thesis
by (metis (no_types, lifting) R.add.m_comm R.minus_equality assms(1)
assms(2) assms(3) ring.ring_simprules(3) ring_hom_closed)
qed
lemma to_fun_minus:
assumes "f ∈ carrier P"
assumes "x ∈ carrier R"
shows "to_fun (⊖⇘P⇙f) x = ⊖ (to_fun f x)"
unfolding to_function_def to_fun_def
using eval_ring_hom[of x] assms
by (simp add: UP_ring ring_hom_uminus)
lemma id_is_hom:
"ring_hom_cring R R (λx. x)"
proof(rule ring_hom_cringI)
show "cring R"
by (simp add: R_cring )
show "cring R"
by (simp add: R_cring )
show "(λx. x) ∈ ring_hom R R"
unfolding ring_hom_def
apply(auto)
done
qed
lemma UP_pre_univ_prop_fact:
"UP_pre_univ_prop R R (λx. x)"
unfolding UP_pre_univ_prop_def
by (simp add: UP_cring_def R_cring id_is_hom)
end
subsection‹to-fun is a Ring Homomorphism from Polynomials to Functions›
context UP_cring
begin
lemma to_fun_is_Fun:
assumes "x ∈ carrier P"
shows "to_fun x ∈ carrier (Fun R)"
apply(rule ring_functions.function_ring_car_memI)
unfolding ring_functions_def apply(simp add: R.ring_axioms)
using to_fun_closed assms apply auto[1]
unfolding to_function_def to_fun_def by auto
lemma to_fun_Fun_mult:
assumes "x ∈ carrier P"
assumes "y ∈ carrier P"
shows "to_fun (x ⊗⇘P⇙ y) = to_fun x ⊗⇘function_ring (carrier R) R⇙ to_fun y"
apply(rule ring_functions.function_ring_car_eqI[of R _ "carrier R"])
apply (simp add: R.ring_axioms ring_functions_def)
apply (simp add: assms(1) assms(2) to_fun_is_Fun)
apply (simp add: R.ring_axioms assms(1) assms(2) ring_functions.fun_mult_closed ring_functions.intro to_fun_is_Fun)
by (simp add: R.ring_axioms assms(1) assms(2) ring_functions.function_mult_eval_car ring_functions.intro to_fun_is_Fun to_fun_mult)
lemma to_fun_Fun_add:
assumes "x ∈ carrier P"
assumes "y ∈ carrier P"
shows "to_fun (x ⊕⇘P⇙ y) = to_fun x ⊕⇘function_ring (carrier R) R⇙ to_fun y"
apply(rule ring_functions.function_ring_car_eqI[of R _ "carrier R"])
apply (simp add: R.ring_axioms ring_functions_def)
apply (simp add: assms(1) assms(2) to_fun_is_Fun)
apply (simp add: R.ring_axioms assms(1) assms(2) ring_functions.fun_add_closed ring_functions.intro to_fun_is_Fun)
by (simp add: R.ring_axioms assms(1) assms(2) ring_functions.fun_add_eval_car ring_functions.intro to_fun_is_Fun to_fun_plus)
lemma to_fun_Fun_one:
"to_fun 𝟭⇘P⇙ = 𝟭⇘Fun R⇙"
apply(rule ring_functions.function_ring_car_eqI[of R _ "carrier R"])
apply (simp add: R.ring_axioms ring_functions_def)
apply (simp add: to_fun_is_Fun)
apply (simp add: R.ring_axioms ring_functions.function_one_closed ring_functions_def)
using P_def R.ring_axioms UP_cring.eval_ring_hom UP_cring.to_fun_eval UP_cring_axioms UP_one_closed ring_functions.function_one_eval ring_functions.intro ring_hom_one
by fastforce
lemma to_fun_Fun_zero:
"to_fun 𝟬⇘P⇙ = 𝟬⇘Fun R⇙"
apply(rule ring_functions.function_ring_car_eqI[of R _ "carrier R"])
apply (simp add: R.ring_axioms ring_functions_def)
apply (simp add: to_fun_is_Fun)
apply (simp add: R.ring_axioms ring_functions.function_zero_closed ring_functions_def)
using P_def R.ring_axioms UP_cring.eval_ring_hom UP_cring.to_fun_eval UP_cring_axioms UP_zero_closed ring_functions.function_zero_eval ring_functions.intro ring_hom_zero
by (metis UP_ring eval_ring_hom)
lemma to_fun_function_ring_hom:
"to_fun ∈ ring_hom P (Fun R)"
apply(rule ring_hom_memI)
using to_fun_is_Fun apply auto[1]
apply (simp add: to_fun_Fun_mult)
apply (simp add: to_fun_Fun_add)
by (simp add: to_fun_Fun_one)
lemma(in UP_cring) to_fun_one:
assumes "a ∈ carrier R"
shows "to_fun 𝟭⇘P⇙ a = 𝟭"
using assms to_fun_Fun_one
by (metis P_def UP_cring.to_fun_eval UP_cring_axioms UP_one_closed eval_ring_hom ring_hom_one)
lemma(in UP_cring) to_fun_zero:
assumes "a ∈ carrier R"
shows "to_fun 𝟬⇘P⇙ a = 𝟬"
by (simp add: assms R.ring_axioms ring_functions.function_zero_eval ring_functions.intro to_fun_Fun_zero)
lemma(in UP_cring) to_fun_nat_pow:
assumes "h ∈ carrier (UP R)"
assumes "a ∈ carrier R"
shows "to_fun (h[^]⇘UP R⇙(n::nat)) a = (to_fun h a)[^]n"
apply(induction n)
using assms to_fun_one
apply (metis P.nat_pow_0 P_def R.nat_pow_0)
using assms to_fun_mult P.nat_pow_closed P_def by auto
lemma(in UP_cring) to_fun_finsum:
assumes "finite (Y::'d set)"
assumes "f ∈ UNIV → carrier (UP R)"
assumes "t ∈ carrier R"
shows "to_fun (finsum (UP R) f Y) t = finsum R (λi. (to_fun (f i) t)) Y"
proof(rule finite.induct[of Y])
show "finite Y"
using assms by blast
show "to_fun (finsum (UP R) f {}) t = (⨁i∈{}. to_fun (f i) t)"
using P.finsum_empty[of f] assms unfolding P_def R.finsum_empty
using P_def to_fun_zero by presburger
show "⋀A a. finite A ⟹
to_fun (finsum (UP R) f A) t = (⨁i∈A. to_fun (f i) t) ⟹ to_fun (finsum (UP R) f (insert a A)) t = (⨁i∈insert a A. to_fun (f i) t)"
proof-
fix A :: "'d set" fix a
assume A: "finite A" "to_fun (finsum (UP R) f A) t = (⨁i∈A. to_fun (f i) t)"
show "to_fun (finsum (UP R) f (insert a A)) t = (⨁i∈insert a A. to_fun (f i) t)"
proof(cases "a ∈ A")
case True
then show ?thesis using A
by (metis insert_absorb)
next
case False
have 0: "finsum (UP R) f (insert a A) = f a ⊕⇘UP R⇙ finsum (UP R) f A"
using A False finsum_insert[of A a f] assms unfolding P_def by blast
have 1: "to_fun (f a ⊕⇘P⇙finsum (UP R) f A ) t = to_fun (f a) t ⊕ to_fun (finsum (UP R) f A) t"
apply(rule to_fun_plus[of "finsum (UP R) f A" "f a" t])
using assms(2) finsum_closed[of f A] A unfolding P_def apply blast
using P_def assms apply blast
using assms by blast
have 2: "to_fun (f a ⊕⇘P⇙finsum (UP R) f A ) t = to_fun (f a) t ⊕ (⨁i∈A. to_fun (f i) t)"
unfolding 1 A by blast
have 3: "(⨁i∈insert a A. to_fun (f i) t) = to_fun (f a) t ⊕ (⨁i∈A. to_fun (f i) t)"
apply(rule R.finsum_insert, rule A, rule False)
using to_fun_closed assms unfolding P_def apply blast
apply(rule to_fun_closed) using assms unfolding P_def apply blast using assms by blast
show ?thesis
unfolding 0 unfolding 3 using 2 unfolding P_def by blast
qed
qed
qed
end
subsection‹Inclusion of a Ring into its Polynomials Ring via Constants›
definition to_polynomial where
"to_polynomial R = (λa. monom (UP R) a 0)"
context UP_cring
begin
abbreviation(input) to_poly where
"to_poly ≡ to_polynomial R"
lemma to_poly_mult_simp:
assumes "b ∈ carrier R"
assumes "f ∈ carrier (UP R)"
shows "(to_polynomial R b) ⊗⇘UP R⇙ f = b ⊙⇘UP R⇙ f"
"f ⊗⇘UP R⇙ (to_polynomial R b) = b ⊙⇘UP R⇙ f"
unfolding to_polynomial_def
using assms P_def monom_mult_is_smult apply auto[1]
using UP_cring.UP_m_comm UP_cring_axioms UP_ring.monom_closed
UP_ring.monom_mult_is_smult UP_ring_axioms assms(1) assms(2)
by fastforce
lemma to_fun_to_poly:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "to_fun (to_poly a) b = a"
unfolding to_function_def to_fun_def to_polynomial_def
by (simp add: UP_pre_univ_prop.eval_const UP_pre_univ_prop_fact assms(1) assms(2))
lemma to_poly_inverse:
assumes "f ∈ carrier P"
assumes "degree f = 0"
shows "f = to_poly (f 0)"
using P_def assms(1) assms(2)
by (metis ltrm_deg_0 to_polynomial_def)
lemma to_poly_closed:
assumes "a ∈ carrier R"
shows "to_poly a ∈ carrier P"
by (metis P_def assms monom_closed to_polynomial_def)
lemma degree_to_poly[simp]:
assumes "a ∈ carrier R"
shows "degree (to_poly a) = 0"
by (metis P_def assms deg_const to_polynomial_def)
lemma to_poly_is_ring_hom:
"to_poly ∈ ring_hom R P"
unfolding to_polynomial_def
unfolding P_def
using UP_ring.const_ring_hom[of R]
UP_ring_axioms by simp
lemma to_poly_add:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "to_poly (a ⊕ b) = to_poly a ⊕⇘P⇙ to_poly b"
by (simp add: assms(1) assms(2) ring_hom_add to_poly_is_ring_hom)
lemma to_poly_mult:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "to_poly (a ⊗ b) = to_poly a ⊗⇘P⇙ to_poly b"
by (simp add: assms(1) assms(2) ring_hom_mult to_poly_is_ring_hom)
lemma to_poly_minus:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "to_poly (a ⊖ b) = to_poly a ⊖⇘P⇙ to_poly b"
by (metis P.minus_eq P_def R.add.inv_closed R.ring_axioms UP_ring.monom_add
UP_ring_axioms assms(1) assms(2) monom_a_inv ring.ring_simprules(14) to_polynomial_def)
lemma to_poly_a_inv:
assumes "a ∈ carrier R"
shows "to_poly (⊖ a) = ⊖⇘P⇙ to_poly a"
by (metis P_def assms monom_a_inv to_polynomial_def)
lemma to_poly_nat_pow:
assumes "a ∈ carrier R"
shows "(to_poly a) [^]⇘P⇙ (n::nat)= to_poly (a[^]n)"
using assms UP_cring UP_cring_axioms UP_cring_def UnivPoly.ring_hom_cringI ring_hom_cring.hom_pow to_poly_is_ring_hom
by fastforce
end
section‹Polynomial Substitution›
definition compose where
"compose R f g = eval R (UP R) (to_polynomial R) g f"
abbreviation(in UP_cring)(input) sub (infixl "of" 70) where
"sub f g ≡ compose R f g"
definition rev_compose where
"rev_compose R = eval R (UP R) (to_polynomial R)"
abbreviation(in UP_cring)(input) rev_sub where
"rev_sub ≡ rev_compose R"
context UP_cring
begin
lemma sub_rev_sub:
"sub f g = rev_sub g f"
unfolding compose_def rev_compose_def
by simp
lemma(in UP_cring) to_poly_UP_pre_univ_prop:
"UP_pre_univ_prop R P to_poly"
proof
show "to_poly ∈ ring_hom R P"
by (simp add: to_poly_is_ring_hom)
qed
lemma rev_sub_is_hom:
assumes "g ∈ carrier P"
shows "rev_sub g ∈ ring_hom P P"
unfolding rev_compose_def
using to_poly_UP_pre_univ_prop assms(1) UP_pre_univ_prop.eval_ring_hom[of R P to_poly g]
unfolding P_def apply auto
done
lemma rev_sub_closed:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
shows "rev_sub q p ∈ carrier P"
using rev_sub_is_hom[of q] assms ring_hom_closed[of "rev_sub q" P P p] by auto
lemma sub_closed:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
shows "sub q p ∈ carrier P"
by (simp add: assms(1) assms(2) rev_sub_closed sub_rev_sub)
lemma rev_sub_add:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "h ∈carrier P"
shows "rev_sub g (f ⊕⇘P⇙ h) = (rev_sub g f) ⊕⇘P⇙ (rev_sub g h)"
using rev_sub_is_hom assms ring_hom_add by fastforce
lemma sub_add:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "h ∈carrier P"
shows "((f ⊕⇘P⇙ h) of g) = ((f of g) ⊕⇘P⇙ (h of g))"
by (simp add: assms(1) assms(2) assms(3) rev_sub_add sub_rev_sub)
lemma rev_sub_mult:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "h ∈carrier P"
shows "rev_sub g (f ⊗⇘P⇙ h) = (rev_sub g f) ⊗⇘P⇙ (rev_sub g h)"
using rev_sub_is_hom assms ring_hom_mult by fastforce
lemma sub_mult:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "h ∈carrier P"
shows "((f ⊗⇘P⇙ h) of g) = ((f of g) ⊗⇘P⇙ (h of g))"
by (simp add: assms(1) assms(2) assms(3) rev_sub_mult sub_rev_sub)
lemma sub_monom:
assumes "g ∈ carrier (UP R)"
assumes "a ∈ carrier R"
shows "sub (monom (UP R) a n) g = to_poly a ⊗⇘UP R⇙ (g[^]⇘UP R⇙ (n::nat))"
"sub (monom (UP R) a n) g = a ⊙⇘UP R⇙ (g[^]⇘UP R⇙ (n::nat))"
apply (simp add: UP_cring.to_poly_UP_pre_univ_prop UP_cring_axioms
UP_pre_univ_prop.eval_monom assms(1) assms(2) Cring_Poly.compose_def)
by (metis P_def UP_cring.to_poly_mult_simp(1) UP_cring_axioms UP_pre_univ_prop.eval_monom
UP_ring assms(1) assms(2) Cring_Poly.compose_def monoid.nat_pow_closed ring_def to_poly_UP_pre_univ_prop)
text‹Subbing into a constant does nothing›
lemma rev_sub_to_poly:
assumes "g ∈ carrier P"
assumes "a ∈ carrier R"
shows "rev_sub g (to_poly a) = to_poly a"
unfolding to_polynomial_def rev_compose_def
using to_poly_UP_pre_univ_prop
unfolding to_polynomial_def
using P_def UP_pre_univ_prop.eval_const assms(1) assms(2) by fastforce
lemma sub_to_poly:
assumes "g ∈ carrier P"
assumes "a ∈ carrier R"
shows "(to_poly a) of g = to_poly a"
by (simp add: assms(1) assms(2) rev_sub_to_poly sub_rev_sub)
lemma sub_const:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "degree f = 0"
shows "f of g = f"
by (metis lcf_closed assms(1) assms(2) assms(3) sub_to_poly to_poly_inverse)
text‹Substitution into a monomial›
lemma monom_sub:
assumes "a ∈ carrier R"
assumes "g ∈ carrier P"
shows "(monom P a n) of g = a ⊙⇘P⇙ g[^]⇘P⇙ n"
unfolding compose_def
using assms UP_pre_univ_prop.eval_monom[of R P to_poly a g n] to_poly_UP_pre_univ_prop
unfolding P_def
using P.nat_pow_closed P_def to_poly_mult_simp(1)
by (simp add: to_poly_mult_simp(1) UP_cring_axioms)
lemma(in UP_cring) cring_sub_monom_bound:
assumes "a ∈ carrier R"
assumes "a ≠𝟬"
assumes "f = monom P a n"
assumes "g ∈ carrier P"
shows "degree (f of g) ≤ n*(degree g)"
proof-
have "f of g = (to_poly a) ⊗⇘P⇙ (g[^]⇘P⇙n)"
unfolding compose_def
using assms UP_pre_univ_prop.eval_monom[of R P to_poly a g] to_poly_UP_pre_univ_prop
unfolding P_def
by blast
then show ?thesis
by (smt (verit) P.nat_pow_closed assms(1) assms(4) cring_pow_deg_bound deg_mult_ring
degree_to_poly le_trans plus_nat.add_0 to_poly_closed)
qed
lemma(in UP_cring) cring_sub_monom:
assumes "a ∈ carrier R"
assumes "a ≠𝟬"
assumes "f = monom P a n"
assumes "g ∈ carrier P"
assumes "a ⊗ (lcf g [^] n) ≠ 𝟬"
shows "degree (f of g) = n*(degree g)"
proof-
have 0: "f of g = (to_poly a) ⊗⇘P⇙ (g[^]⇘P⇙n)"
unfolding compose_def
using assms UP_pre_univ_prop.eval_monom[of R P to_poly a g] to_poly_UP_pre_univ_prop
unfolding P_def
by blast
have 1: "lcf (to_poly a) ⊗ lcf (g [^]⇘P⇙ n) ≠ 𝟬"
using assms
by (smt (verit) P.nat_pow_closed P_def R.nat_pow_closed R.r_null cring_pow_ltrm lcf_closed lcf_ltrm lcf_monom monom_pow to_polynomial_def)
then show ?thesis
using 0 1 assms cring_pow_deg[of g n] cring_deg_mult[of "to_poly a" "g[^]⇘P⇙n"]
by (metis P.nat_pow_closed R.r_null add.right_neutral degree_to_poly to_poly_closed)
qed
lemma(in UP_domain) sub_monom:
assumes "a ∈ carrier R"
assumes "a ≠𝟬"
assumes "f = monom P a n"
assumes "g ∈ carrier P"
shows "degree (f of g) = n*(degree g)"
proof-
have "f of g = (to_poly a) ⊗⇘P⇙ (g[^]⇘P⇙n)"
unfolding compose_def
using assms UP_pre_univ_prop.eval_monom[of R P to_poly a g] to_poly_UP_pre_univ_prop
unfolding P_def
by blast
then show ?thesis using deg_pow deg_mult
by (metis P.nat_pow_closed P_def assms(1) assms(2)
assms(4) deg_smult monom_mult_is_smult to_polynomial_def)
qed
text‹Subbing a constant into a polynomial yields a constant›
lemma sub_in_const:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "degree g = 0"
shows "degree (f of g) = 0"
proof-
have "⋀n. (⋀p. p ∈ carrier P ⟹ degree p ≤ n ⟹ degree (p of g) = 0)"
proof-
fix n
show "⋀p. p ∈ carrier P ⟹ degree p ≤ n ⟹ degree (p of g) = 0"
proof(induction n)
case 0
then show ?case
by (simp add: assms(1) sub_const)
next
case (Suc n)
fix n
assume IH: "⋀p. p ∈ carrier P ⟹ degree p ≤ n ⟹ degree (p of g) = 0"
show "⋀p. p ∈ carrier P ⟹ degree p ≤ (Suc n) ⟹ degree (p of g) = 0"
proof-
fix p
assume A0: "p ∈ carrier P"
assume A1: "degree p ≤ (Suc n)"
show "degree (p of g) = 0"
proof(cases "degree p < Suc n")
case True
then show ?thesis using IH
using A0 by auto
next
case False
then have D: "degree p = Suc n"
by (simp add: A1 nat_less_le)
show ?thesis
proof-
have P0: "degree ((trunc p) of g) = 0" using IH
by (metis A0 D less_Suc_eq_le trunc_degree trunc_closed zero_less_Suc)
have P1: "degree ((ltrm p) of g) = 0"
proof-
obtain a n where an_def: "ltrm p = monom P a n ∧ a ∈ carrier R"
unfolding leading_term_def
using A0 P_def cfs_closed by blast
obtain b where b_def: "g = monom P b 0 ∧ b ∈ carrier R"
using assms deg_zero_impl_monom coeff_closed
by blast
have 0: " monom P b 0 [^]⇘P⇙ n = monom P (b[^]n) 0"
apply(induction n)
apply fastforce[1]
proof- fix n::nat assume IH: "monom P b 0 [^]⇘P⇙ n = monom P (b [^] n) 0"
have "monom P b 0 [^]⇘P⇙ Suc n = (monom P (b[^]n) 0) ⊗⇘P⇙ monom P b 0"
using IH by simp
then have "monom P b 0 [^]⇘P⇙ Suc n = (monom P ((b[^]n)⊗b) 0)"
using b_def
by (simp add: monom_mult_is_smult monom_mult_smult)
then show "monom P b 0 [^]⇘P⇙ Suc n = monom P (b [^] Suc n) 0 "
by simp
qed
then have 0: "a ⊙⇘P⇙ monom P b 0 [^]⇘P⇙ n = monom P (a ⊗ b[^]n) 0"
by (simp add: an_def b_def monom_mult_smult)
then show ?thesis using monom_sub[of a "monom P b 0" n] assms an_def
by (simp add: ‹⟦a ∈ carrier R; monom P b 0 ∈ carrier P⟧ ⟹ monom P a n of monom P b 0 = a ⊙⇘P⇙ monom P b 0 [^]⇘P⇙ n› b_def)
qed
have P2: "p of g = (trunc p of g) ⊕⇘P⇙ ((ltrm p) of g)"
by (metis A0 assms(1) ltrm_closed sub_add trunc_simps(1) trunc_closed)
then show ?thesis
using P0 P1 P2 deg_add[of "trunc p of g" "ltrm p of g"]
by (metis A0 assms(1) le_0_eq ltrm_closed max_0R sub_closed trunc_closed)
qed
qed
qed
qed
qed
then show ?thesis
using assms(2) by blast
qed
lemma (in UP_cring) cring_sub_deg_bound:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
shows "degree (f of g) ≤ degree f * degree g"
proof-
have "⋀n. ⋀ p. p ∈ carrier P ⟹ (degree p) ≤ n ⟹ degree (p of g) ≤ degree p * degree g"
proof-
fix n::nat
show "⋀ p. p ∈ carrier P ⟹ (degree p) ≤ n ⟹ degree (p of g) ≤ degree p * degree g"
proof(induction n)
case 0
then have B0: "degree p = 0" by auto
then show ?case using sub_const[of g p]
by (simp add: "0.prems"(1) assms(1))
next
case (Suc n)
fix n
assume IH: "(⋀p. p ∈ carrier P ⟹ degree p ≤ n ⟹ degree (p of g) ≤ degree p * degree g)"
show " p ∈ carrier P ⟹ degree p ≤ Suc n ⟹ degree (p of g) ≤ degree p * degree g"
proof-
assume A0: "p ∈ carrier P"
assume A1: "degree p ≤ Suc n"
show ?thesis
proof(cases "degree p < Suc n")
case True
then show ?thesis using IH
by (simp add: A0)
next
case False
then have D: "degree p = Suc n"
using A1 by auto
have P0: "(p of g) = ((trunc p) of g) ⊕⇘P⇙ ((ltrm p) of g)"
by (metis A0 assms(1) ltrm_closed sub_add trunc_simps(1) trunc_closed)
have P1: "degree ((trunc p) of g) ≤ (degree (trunc p))*(degree g)"
using IH by (metis A0 D less_Suc_eq_le trunc_degree trunc_closed zero_less_Suc)
have P2: "degree ((ltrm p) of g) ≤ (degree p) * degree g"
using A0 D P_def UP_cring_axioms assms(1)
by (metis False cfs_closed coeff_simp cring_sub_monom_bound deg_zero lcoeff_nonzero2 less_Suc_eq_0_disj)
then show ?thesis
proof(cases "degree g = 0")
case True
then show ?thesis
by (simp add: Suc(2) assms(1) sub_in_const)
next
case F: False
then show ?thesis
proof-
have P3: "degree ((trunc p) of g) ≤ n*degree g"
using A0 False D P1 P2 IH[of "trunc p"] trunc_degree[of p]
proof -
{ assume "degree (trunc p) < degree p"
then have "degree (trunc p) ≤ n"
using D by auto
then have ?thesis
by (meson P1 le_trans mult_le_cancel2) }
then show ?thesis
by (metis (full_types) A0 D Suc_mult_le_cancel1 nat_mult_le_cancel_disj trunc_degree)
qed
then have P3': "degree ((trunc p) of g) < (degree p)*degree g"
using F D by auto
have P4: "degree (ltrm p of g) ≤ (degree p)*degree g"
using cring_sub_monom_bound D P2
by auto
then show ?thesis
using D P0 P1 P3 P4 A0 P3' assms(1) bound_deg_sum less_imp_le_nat
ltrm_closed sub_closed trunc_closed
by metis
qed
qed
qed
qed
qed
qed
then show ?thesis
using assms(2) by blast
qed
lemma (in UP_cring) cring_sub_deg:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "lcf f ⊗ (lcf g [^] (degree f)) ≠ 𝟬"
shows "degree (f of g) = degree f * degree g"
proof-
have 0: "f of g = (trunc f of g) ⊕⇘P⇙ ((ltrm f) of g)"
by (metis assms(1) assms(2) ltrm_closed rev_sub_add sub_rev_sub trunc_simps(1) trunc_closed)
have 1: "lcf f ≠ 𝟬"
using assms cring.cring_simprules(26) lcf_closed
by auto
have 2: "degree ((ltrm f) of g) = degree f * degree g"
using 0 1 assms cring_sub_monom[of "lcf f" "ltrm f" "degree f" g] lcf_closed lcf_ltrm
by blast
show ?thesis
apply(cases "degree f = 0")
apply (simp add: assms(1) assms(2))
apply(cases "degree g = 0")
apply (simp add: assms(1) assms(2) sub_in_const)
using 0 1 assms cring_sub_deg_bound[of g "trunc f"] trunc_degree[of f]
using sub_const apply auto[1]
apply(cases "degree g = 0")
using 0 1 assms cring_sub_deg_bound[of g "trunc f"] trunc_degree[of f]
using sub_in_const apply fastforce
unfolding 0 using 1 2
by (smt (verit) "0" ltrm_closed ‹⟦f ∈ carrier P; 0 < deg R f⟧ ⟹ deg R (Cring_Poly.truncate R f) < deg R f›
assms(1) assms(2) cring_sub_deg_bound degree_of_sum_diff_degree equal_deg_sum
le_eq_less_or_eq mult_less_cancel2 nat_neq_iff neq0_conv sub_closed trunc_closed)
qed
lemma (in UP_domain) sub_deg0:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "g ≠ 𝟬⇘P⇙"
assumes "f ≠ 𝟬⇘P⇙"
shows "degree (f of g) = degree f * degree g"
proof-
have "⋀n. ⋀ p. p ∈ carrier P ⟹ (degree p) ≤ n ⟹ degree (p of g) = degree p * degree g"
proof-
fix n::nat
show "⋀ p. p ∈ carrier P ⟹ (degree p) ≤ n ⟹ degree (p of g) = degree p * degree g"
proof(induction n)
case 0
then have B0: "degree p = 0" by auto
then show ?case using sub_const[of g p]
by (simp add: "0.prems"(1) assms(1))
next
case (Suc n)
fix n
assume IH: "(⋀p. p ∈ carrier P ⟹ degree p ≤ n ⟹ degree (p of g) = degree p * degree g)"
show " p ∈ carrier P ⟹ degree p ≤ Suc n ⟹ degree (p of g) = degree p * degree g"
proof-
assume A0: "p ∈ carrier P"
assume A1: "degree p ≤ Suc n"
show ?thesis
proof(cases "degree p < Suc n")
case True
then show ?thesis using IH
by (simp add: A0)
next
case False
then have D: "degree p = Suc n"
using A1 by auto
have P0: "(p of g) = ((trunc p) of g) ⊕⇘P⇙ ((ltrm p) of g)"
by (metis A0 assms(1) ltrm_closed sub_add trunc_simps(1) trunc_closed)
have P1: "degree ((trunc p) of g) = (degree (trunc p))*(degree g)"
using IH by (metis A0 D less_Suc_eq_le trunc_degree trunc_closed zero_less_Suc)
have P2: "degree ((ltrm p) of g) = (degree p) * degree g"
using A0 D P_def UP_domain.sub_monom UP_cring_axioms assms(1)
by (metis False UP_domain_axioms UP_ring.coeff_simp UP_ring.lcoeff_nonzero2 UP_ring_axioms cfs_closed deg_nzero_nzero less_Suc_eq_0_disj)
then show ?thesis
proof(cases "degree g = 0")
case True
then show ?thesis
by (simp add: Suc(2) assms(1) sub_in_const)
next
case False
then show ?thesis
proof-
have P3: "degree ((trunc p) of g) < degree ((ltrm p) of g)"
using False D P1 P2
by (metis (no_types, lifting) A0 mult.commute mult_right_cancel
nat_less_le nat_mult_le_cancel_disj trunc_degree zero_less_Suc)
then show ?thesis
by (simp add: A0 ltrm_closed P0 P2 assms(1) equal_deg_sum sub_closed trunc_closed)
qed
qed
qed
qed
qed
qed
then show ?thesis
using assms(2) by blast
qed
lemma(in UP_domain) sub_deg:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "g ≠ 𝟬⇘P⇙"
shows "degree (f of g) = degree f * degree g"
proof(cases "f = 𝟬⇘P⇙")
case True
then show ?thesis
using assms(1) sub_const by auto
next
case False
then show ?thesis
by (simp add: assms(1) assms(2) assms(3) sub_deg0)
qed
lemma(in UP_cring) cring_ltrm_sub:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "degree g > 0"
assumes "lcf f ⊗ (lcf g [^] (degree f)) ≠ 𝟬"
shows "ltrm (f of g) = ltrm ((ltrm f) of g)"
proof-
have P0: "degree (f of g) = degree ((ltrm f) of g)"
using assms(1) assms(2) assms(4) cring_sub_deg lcf_eq ltrm_closed deg_ltrm
by auto
have P1: "f of g = ((trunc f) of g) ⊕⇘P⇙((ltrm f) of g)"
by (metis assms(1) assms(2) ltrm_closed rev_sub_add sub_rev_sub trunc_simps(1) trunc_closed)
then show ?thesis
proof(cases "degree f = 0")
case True
then show ?thesis
using ltrm_deg_0 assms(2) by auto
next
case False
have P2: "degree (f of g) = degree f * degree g"
by (simp add: assms(1) assms(2) assms(4) cring_sub_deg)
then have P3: "degree ((trunc f) of g) < degree ((ltrm f) of g)"
using False P0 P1 P_def UP_cring.sub_closed trunc_closed UP_cring_axioms
UP_ring.degree_of_sum_diff_degree UP_ring.ltrm_closed UP_ring_axioms assms(1)
assms(2) assms(4) cring_sub_deg_bound le_antisym less_imp_le_nat less_nat_zero_code
mult_right_le_imp_le nat_neq_iff trunc_degree
by (smt (verit) assms(3))
then show ?thesis using P0 P1 P2
by (metis (no_types, lifting) ltrm_closed ltrm_of_sum_diff_degree P.add.m_comm assms(1) assms(2) sub_closed trunc_closed)
qed
qed
lemma(in UP_domain) ltrm_sub:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "degree g > 0"
shows "ltrm (f of g) = ltrm ((ltrm f) of g)"
proof-
have P0: "degree (f of g) = degree ((ltrm f) of g)"
using sub_deg
by (metis ltrm_closed assms(1) assms(2) assms(3) deg_zero deg_ltrm nat_neq_iff)
have P1: "f of g = ((trunc f) of g) ⊕⇘P⇙((ltrm f) of g)"
by (metis assms(1) assms(2) ltrm_closed rev_sub_add sub_rev_sub trunc_simps(1) trunc_closed)
then show ?thesis
proof(cases "degree f = 0")
case True
then show ?thesis
using ltrm_deg_0 assms(2) by auto
next
case False
then have P2: "degree ((trunc f) of g) < degree ((ltrm f) of g)"
using sub_deg
by (metis (no_types, lifting) ltrm_closed assms(1) assms(2) assms(3) deg_zero
deg_ltrm mult_less_cancel2 neq0_conv trunc_closed trunc_degree)
then show ?thesis
using P0 P1 P2
by (metis (no_types, lifting) ltrm_closed ltrm_of_sum_diff_degree P.add.m_comm assms(1) assms(2) sub_closed trunc_closed)
qed
qed
lemma(in UP_cring) cring_lcf_of_sub_in_ltrm:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "degree f = n"
assumes "degree g > 0"
assumes "(lcf f) ⊗ ((lcf g)[^]n) ≠𝟬"
shows "lcf ((ltrm f) of g) = (lcf f) ⊗ ((lcf g)[^]n)"
by (metis (no_types, lifting) P.nat_pow_closed P_def R.r_null UP_cring.monom_sub UP_cring_axioms
assms(1) assms(2) assms(3) assms(5) cfs_closed cring_lcf_pow cring_lcf_scalar_mult)
lemma(in UP_domain) lcf_of_sub_in_ltrm:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "degree f = n"
assumes "degree g > 0"
shows "lcf ((ltrm f) of g) = (lcf f) ⊗ ((lcf g)[^]n)"
proof(cases "degree f = 0")
case True
then show ?thesis
using ltrm_deg_0 assms(1) assms(2) assms(3) cfs_closed
by (simp add: sub_const)
next
case False
then show ?thesis
proof-
have P0: "(ltrm f) of g = (to_poly (lcf f)) ⊗⇘P⇙ (g[^]⇘P⇙n)"
unfolding compose_def
using assms UP_pre_univ_prop.eval_monom[of R P to_poly "(lcf f)" g n] to_poly_UP_pre_univ_prop
unfolding P_def
using P_def cfs_closed by blast
have P1: "(ltrm f) of g = (lcf f) ⊙⇘P⇙(g[^]⇘P⇙n)"
using P0 P.nat_pow_closed
by (simp add: assms(1) assms(2) assms(3) cfs_closed monom_sub)
have P2: "ltrm ((ltrm f) of g) = (ltrm (to_poly (lcf f))) ⊗⇘P⇙ (ltrm (g[^]⇘P⇙n))"
using P0 ltrm_mult P.nat_pow_closed P_def assms(1) assms(2)
to_poly_closed
by (simp add: cfs_closed)
have P3: "ltrm ((ltrm f) of g) = (to_poly (lcf f)) ⊗⇘P⇙ (ltrm (g[^]⇘P⇙n))"
using P2 ltrm_deg_0 assms(2) to_poly_closed
by (simp add: cfs_closed)
have P4: "ltrm ((ltrm f) of g) = (lcf f) ⊙⇘P⇙ ((ltrm g)[^]⇘P⇙n)"
using P.nat_pow_closed P1 P_def assms(1) assms(2) ltrm_pow0 ltrm_smult
by (simp add: cfs_closed)
have P5: "lcf ((ltrm f) of g) = (lcf f) ⊗ (lcf ((ltrm g)[^]⇘P⇙n))"
using lcf_scalar_mult P4 by (metis P.nat_pow_closed P1 cfs_closed
UP_smult_closed assms(1) assms(2) assms(3) lcf_eq ltrm_closed sub_rev_sub)
show ?thesis
using P5 ltrm_pow lcf_pow assms(1) lcf_eq ltrm_closed by presburger
qed
qed
lemma(in UP_cring) cring_ltrm_of_sub_in_ltrm:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "degree f = n"
assumes "degree g > 0"
assumes "(lcf f) ⊗ ((lcf g)[^]n) ≠𝟬"
shows "ltrm ((ltrm f) of g) = (lcf f) ⊙⇘P⇙ ((ltrm g)[^]⇘P⇙n)"
by (smt (verit) lcf_eq ltrm_closed R.nat_pow_closed R.r_null assms(1) assms(2) assms(3)
assms(4) assms(5) cfs_closed cring_lcf_of_sub_in_ltrm cring_lcf_pow cring_pow_ltrm
cring_pow_deg cring_sub_deg deg_zero deg_ltrm monom_mult_smult neq0_conv)
lemma(in UP_domain) ltrm_of_sub_in_ltrm:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "degree f = n"
assumes "degree g > 0"
shows "ltrm ((ltrm f) of g) = (lcf f) ⊙⇘P⇙ ((ltrm g)[^]⇘P⇙n)"
using assms(1) assms(2) assms(3) lcf_closed ltrm_pow0 ltrm_smult monom_sub by force
text‹formula for the leading term of a composition ›
lemma(in UP_domain) cring_ltrm_of_sub:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "degree f = n"
assumes "degree g > 0"
assumes "(lcf f) ⊗ ((lcf g)[^]n) ≠𝟬"
shows "ltrm (f of g) = (lcf f) ⊙⇘P⇙ ((ltrm g)[^]⇘P⇙n)"
using ltrm_of_sub_in_ltrm ltrm_sub assms(1) assms(2) assms(3) assms(4) by presburger
lemma(in UP_domain) ltrm_of_sub:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "degree f = n"
assumes "degree g > 0"
shows "ltrm (f of g) = (lcf f) ⊙⇘P⇙ ((ltrm g)[^]⇘P⇙n)"
using ltrm_of_sub_in_ltrm ltrm_sub assms(1) assms(2) assms(3) assms(4) by presburger
text‹subtitution is associative›
lemma sub_assoc_monom:
assumes "f ∈ carrier P"
assumes "q ∈ carrier P"
assumes "r ∈ carrier P"
shows "(ltrm f) of (q of r) = ((ltrm f) of q) of r"
proof-
obtain n where n_def: "n = degree f"
by simp
obtain a where a_def: "a ∈ carrier R ∧ (ltrm f) = monom P a n"
using assms(1) cfs_closed n_def by blast
have LHS: "(ltrm f) of (q of r) = a ⊙⇘P⇙ (q of r)[^]⇘P⇙ n"
by (metis P.nat_pow_closed P_def UP_pre_univ_prop.eval_monom a_def assms(2)
assms(3) compose_def monom_mult_is_smult sub_closed to_poly_UP_pre_univ_prop to_polynomial_def)
have RHS0: "((ltrm f) of q) of r = (a ⊙⇘P⇙ q[^]⇘P⇙ n)of r"
by (metis P.nat_pow_closed P_def UP_pre_univ_prop.eval_monom a_def
assms(2) compose_def monom_mult_is_smult to_poly_UP_pre_univ_prop to_polynomial_def)
have RHS1: "((ltrm f) of q) of r = ((to_poly a) ⊗⇘P⇙ q[^]⇘P⇙ n)of r"
using RHS0 by (metis P.nat_pow_closed P_def a_def
assms(2) monom_mult_is_smult to_polynomial_def)
have RHS2: "((ltrm f) of q) of r = ((to_poly a) of r) ⊗⇘P⇙ (q[^]⇘P⇙ n of r)"
using RHS1 a_def assms(2) assms(3) sub_mult to_poly_closed by auto
have RHS3: "((ltrm f) of q) of r = (to_poly a) ⊗⇘P⇙ (q[^]⇘P⇙ n of r)"
using RHS2 a_def assms(3) sub_to_poly by auto
have RHS4: "((ltrm f) of q) of r = a ⊙⇘P⇙ ((q[^]⇘P⇙ n)of r)"
using RHS3
by (metis P.nat_pow_closed P_def a_def assms(2) assms(3)
monom_mult_is_smult sub_closed to_polynomial_def)
have "(q of r)[^]⇘P⇙ n = ((q[^]⇘P⇙ n)of r)"
apply(induction n)
apply (metis Group.nat_pow_0 P.ring_simprules(6) assms(3) deg_one sub_const)
by (simp add: assms(2) assms(3) sub_mult)
then show ?thesis using RHS4 LHS by simp
qed
lemma sub_assoc:
assumes "f ∈ carrier P"
assumes "q ∈ carrier P"
assumes "r ∈ carrier P"
shows "f of (q of r) = (f of q) of r"
proof-
have "⋀ n. ⋀ p. p ∈ carrier P ⟹ degree p ≤ n ⟹ p of (q of r) = (p of q) of r"
proof-
fix n
show "⋀ p. p ∈ carrier P ⟹ degree p ≤ n ⟹ p of (q of r) = (p of q) of r"
proof(induction n)
case 0
then have deg_p: "degree p = 0"
by blast
then have B0: "p of (q of r) = p"
using sub_const[of "q of r" p] assms "0.prems"(1) sub_closed by blast
have B1: "(p of q) of r = p"
proof-
have p0: "p of q = p"
using deg_p 0 assms(2)
by (simp add: P_def UP_cring.sub_const UP_cring_axioms)
show ?thesis
unfolding p0 using deg_p 0 assms(3)
by (simp add: P_def UP_cring.sub_const UP_cring_axioms)
qed
then show "p of (q of r) = (p of q) of r" using B0 B1 by auto
next
case (Suc n)
fix n
assume IH: "⋀ p. p ∈ carrier P ⟹ degree p ≤ n ⟹ p of (q of r) = (p of q) of r"
then show "⋀ p. p ∈ carrier P ⟹ degree p ≤ Suc n ⟹ p of (q of r) = (p of q) of r"
proof-
fix p
assume A0: " p ∈ carrier P "
assume A1: "degree p ≤ Suc n"
show "p of (q of r) = (p of q) of r"
proof(cases "degree p < Suc n")
case True
then show ?thesis using A0 A1 IH by auto
next
case False
then have "degree p = Suc n"
using A1 by auto
have I0: "p of (q of r) = ((trunc p) ⊕⇘P⇙ (ltrm p)) of (q of r)"
using A0 trunc_simps(1) by auto
have I1: "p of (q of r) = ((trunc p) of (q of r)) ⊕⇘P⇙ ((ltrm p) of (q of r))"
using I0 sub_add
by (simp add: A0 assms(2) assms(3) ltrm_closed rev_sub_closed sub_rev_sub trunc_closed)
have I2: "p of (q of r) = (((trunc p) of q) of r) ⊕⇘P⇙ (((ltrm p) of q) of r)"
using IH[of "trunc p"] sub_assoc_monom[of p q r]
by (metis A0 I1 ‹degree p = Suc n› assms(2) assms(3)
less_Suc_eq_le trunc_degree trunc_closed zero_less_Suc)
have I3: "p of (q of r) = (((trunc p) of q) ⊕⇘P⇙ ((ltrm p) of q)) of r"
using sub_add trunc_simps(1) assms
by (simp add: A0 I2 ltrm_closed sub_closed trunc_closed)
have I4: "p of (q of r) = (((trunc p)⊕⇘P⇙(ltrm p)) of q) of r"
using sub_add trunc_simps(1) assms
by (simp add: trunc_simps(1) A0 I3 ltrm_closed trunc_closed)
then show ?thesis
using A0 trunc_simps(1) by auto
qed
qed
qed
qed
then show ?thesis
using assms(1) by blast
qed
lemma sub_smult:
assumes "f ∈ carrier P"
assumes "q ∈ carrier P"
assumes "a ∈ carrier R"
shows "(a⊙⇘P⇙f ) of q = a⊙⇘P⇙(f of q)"
proof-
have "(a⊙⇘P⇙f ) of q = ((to_poly a) ⊗⇘P⇙f) of q"
using assms by (metis P_def monom_mult_is_smult to_polynomial_def)
then have "(a⊙⇘P⇙f ) of q = ((to_poly a) of q) ⊗⇘P⇙(f of q)"
by (simp add: assms(1) assms(2) assms(3) sub_mult to_poly_closed)
then have "(a⊙⇘P⇙f ) of q = (to_poly a) ⊗⇘P⇙(f of q)"
by (simp add: assms(2) assms(3) sub_to_poly)
then show ?thesis
by (metis P_def assms(1) assms(2) assms(3)
monom_mult_is_smult sub_closed to_polynomial_def)
qed
lemma to_fun_sub_monom:
assumes "is_UP_monom f"
assumes "g ∈ carrier P"
assumes "a ∈ carrier R"
shows "to_fun (f of g) a = to_fun f (to_fun g a)"
proof-
obtain b n where b_def: "b ∈ carrier R ∧ f = monom P b n"
using assms unfolding is_UP_monom_def
using P_def cfs_closed by blast
then have P0: "f of g = b ⊙⇘P⇙ (g[^]⇘P⇙n)"
using b_def assms(2) monom_sub by blast
have P1: "UP_pre_univ_prop R R (λx. x)"
by (simp add: UP_pre_univ_prop_fact)
then have P2: "to_fun f (to_fun g a) = b ⊗((to_fun g a)[^]n)"
using P1 to_fun_eval[of f "to_fun g a"] P_def UP_pre_univ_prop.eval_monom assms(1)
assms(2) assms(3) b_def is_UP_monomE(1) to_fun_closed
by force
have P3: "to_fun (monom P b n of g) a = b ⊗((to_fun g a)[^]n)"
proof-
have 0: "to_fun (monom P b n of g) a = eval R R (λx. x) a (b ⊙⇘P⇙ (g[^]⇘P⇙n) )"
using UP_pre_univ_prop.eval_monom[of R "(UP R)" to_poly b g n]
P_def assms(2) b_def to_poly_UP_pre_univ_prop to_fun_eval P0
by (metis assms(3) monom_closed sub_closed)
have 1: "to_fun (monom P b n of g) a = (eval R R (λx. x) a (to_poly b)) ⊗ ( eval R R (λx. x) a ( g [^]⇘UP R⇙ n ))"
using 0 eval_ring_hom
by (metis P.nat_pow_closed P0 P_def assms(2) assms(3) b_def monom_mult_is_smult to_fun_eval to_fun_mult to_poly_closed to_polynomial_def)
have 2: "to_fun (monom P b n of g) a = b ⊗ ( eval R R (λx. x) a ( g [^]⇘UP R⇙ n ))"
using 1 assms(3) b_def to_fun_eval to_fun_to_poly to_poly_closed by auto
then show ?thesis
unfolding to_function_def to_fun_def
using eval_ring_hom P_def UP_pre_univ_prop.ring_homD UP_pre_univ_prop_fact
assms(2) assms(3) ring_hom_cring.hom_pow by fastforce
qed
then show ?thesis
using b_def P2 by auto
qed
lemma to_fun_sub:
assumes "g ∈ carrier P"
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "to_fun (f of g) a = (to_fun f) (to_fun g a)"
proof(rule poly_induct2[of f])
show "f ∈ carrier P"
using assms by auto
show "⋀p. p ∈ carrier P ⟹ degree p = 0 ⟹ to_fun (p of g) a = to_fun p (to_fun g a)"
proof-
fix p
assume A0: "p ∈ carrier P"
assume A1: "degree p = 0"
then have P0: "degree (p of g) = 0"
by (simp add: A0 assms(1) sub_const)
then obtain b where b_def: "p of g = to_poly b ∧ b ∈ carrier R"
using A0 A1 cfs_closed assms(1) to_poly_inverse
by (meson sub_closed)
then have "to_fun (p of g) a = b"
by (simp add: assms(3) to_fun_to_poly)
have "p of g = p"
using A0 A1 P_def sub_const UP_cring_axioms assms(1) by blast
then have P1: "p = to_poly b"
using b_def by auto
have "to_fun g a ∈ carrier R"
using assms
by (simp add: to_fun_closed)
then show "to_fun (p of g) a = to_fun p (to_fun g a)"
using P1 ‹to_fun (p of g) a = b› b_def
by (simp add: to_fun_to_poly)
qed
show "⋀p. 0 < degree p ⟹ p ∈ carrier P ⟹
to_fun (trunc p of g) a = to_fun (trunc p) (to_fun g a) ⟹
to_fun (p of g) a = to_fun p (to_fun g a)"
proof-
fix p
assume A0: "0 < degree p"
assume A1: " p ∈ carrier P"
assume A2: "to_fun (trunc p of g) a = to_fun (trunc p) (to_fun g a)"
show "to_fun (p of g) a = to_fun p (to_fun g a)"
proof-
have "p of g = (trunc p) of g ⊕⇘P⇙ (ltrm p) of g"
by (metis A1 assms(1) ltrm_closed sub_add trunc_simps(1) trunc_closed)
then have "to_fun (p of g) a = to_fun ((trunc p) of g) a ⊕ (to_fun ((ltrm p) of g) a)"
by (simp add: A1 assms(1) assms(3) to_fun_plus ltrm_closed sub_closed trunc_closed)
then have 0: "to_fun (p of g) a = to_fun (trunc p) (to_fun g a) ⊕ (to_fun ((ltrm p) of g) a)"
by (simp add: A2)
have "(to_fun ((ltrm p) of g) a) = to_fun (ltrm p) (to_fun g a)"
using to_fun_sub_monom
by (simp add: A1 assms(1) assms(3) ltrm_is_UP_monom)
then have "to_fun (p of g) a = to_fun (trunc p) (to_fun g a) ⊕ to_fun (ltrm p) (to_fun g a)"
using 0 by auto
then show ?thesis
by (metis A1 assms(1) assms(3) to_fun_closed to_fun_plus ltrm_closed trunc_simps(1) trunc_closed)
qed
qed
qed
end
text‹More material on constant terms and constant coefficients›
context UP_cring
begin
lemma to_fun_ctrm:
assumes "f ∈ carrier P"
assumes "b ∈ carrier R"
shows "to_fun (ctrm f) b = (f 0)"
using assms
by (metis ctrm_degree ctrm_is_poly lcf_monom(2) P_def cfs_closed to_fun_to_poly to_poly_inverse)
lemma to_fun_smult:
assumes "f ∈ carrier P"
assumes "b ∈ carrier R"
assumes "c ∈ carrier R"
shows "to_fun (c ⊙⇘P⇙ f) b = c ⊗(to_fun f b)"
proof-
have "(c ⊙⇘P⇙ f) = (to_poly c) ⊗⇘P⇙ f"
by (metis P_def assms(1) assms(3) monom_mult_is_smult to_polynomial_def)
then have "to_fun (c ⊙⇘P⇙ f) b = to_fun (to_poly c) b ⊗ to_fun f b"
by (simp add: assms(1) assms(2) assms(3) to_fun_mult to_poly_closed)
then show ?thesis
by (simp add: assms(2) assms(3) to_fun_to_poly)
qed
lemma to_fun_monom:
assumes "c ∈ carrier R"
assumes "x ∈ carrier R"
shows "to_fun (monom P c n) x = c ⊗ x [^] n"
by (smt (verit) P_def R.m_comm R.nat_pow_closed UP_cring.to_poly_nat_pow UP_cring_axioms assms(1)
assms(2) monom_is_UP_monom(1) sub_monom(1) to_fun_smult to_fun_sub_monom to_fun_to_poly
to_poly_closed to_poly_mult_simp(2))
lemma zcf_monom:
assumes "a ∈ carrier R"
shows "zcf (monom P a n) = to_fun (monom P a n) 𝟬"
using to_fun_monom unfolding zcf_def
by (simp add: R.nat_pow_zero assms cfs_monom)
lemma zcf_to_fun:
assumes "p ∈ carrier P"
shows "zcf p = to_fun p 𝟬"
apply(rule poly_induct3[of p])
apply (simp add: assms)
using R.zero_closed zcf_add to_fun_plus apply presburger
using zcf_monom by blast
lemma zcf_to_poly[simp]:
assumes "a ∈ carrier R"
shows "zcf (to_poly a) = a"
by (metis assms cfs_closed degree_to_poly to_fun_to_poly to_poly_inverse to_poly_closed zcf_def)
lemma zcf_ltrm_mult:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "degree p > 0"
shows "zcf((ltrm p) ⊗⇘P⇙ q) = 𝟬"
using zcf_to_fun[of "ltrm p ⊗⇘P⇙ q" ]
by (metis ltrm_closed P.l_null P.m_closed R.zero_closed UP_zero_closed zcf_to_fun
zcf_zero assms(1) assms(2) assms(3) coeff_ltrm to_fun_mult)
lemma zcf_mult:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
shows "zcf(p ⊗⇘P⇙ q) = (zcf p) ⊗ (zcf q)"
using zcf_to_fun[of " p ⊗⇘P⇙ q" ] zcf_to_fun[of "p" ] zcf_to_fun[of "q" ] to_fun_mult[of q p 𝟬]
by (simp add: assms(1) assms(2))
lemma zcf_is_ring_hom:
"zcf∈ ring_hom P R"
apply(rule ring_hom_memI)
using zcf_mult zcf_add
apply (simp add: P_def UP_ring.cfs_closed UP_ring_axioms zcf_def)
apply (simp add: zcf_mult)
using zcf_add apply auto[1]
by simp
lemma ctrm_is_ring_hom:
"ctrm ∈ ring_hom P P"
apply(rule ring_hom_memI)
apply (simp add: ctrm_is_poly)
apply (metis zcf_def zcf_mult cfs_closed monom_mult zero_eq_add_iff_both_eq_0)
using cfs_add[of _ _ 0]
apply (simp add: cfs_closed)
by auto
section‹Describing the Image of (UP R) in the Ring of Functions from R to R›
lemma to_fun_diff:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "a ∈ carrier R"
shows "to_fun (p ⊖⇘P⇙ q) a = to_fun p a ⊖ to_fun q a"
using to_fun_plus[of "⊖⇘P⇙ q" p a]
by (simp add: P.minus_eq R.minus_eq assms(1) assms(2) assms(3) to_fun_minus)
lemma to_fun_const:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "to_fun (monom P a 0) b = a"
by (metis lcf_monom(2) P_def UP_cring.to_fun_ctrm UP_cring_axioms assms(1) assms(2) deg_const monom_closed)
lemma to_fun_monic_monom:
assumes "b ∈ carrier R"
shows "to_fun (monom P 𝟭 n) b = b[^]n"
by (simp add: assms to_fun_monom)
text‹Constant polynomials map to constant polynomials›
lemma const_to_constant:
assumes "a ∈ carrier R"
shows "to_fun (monom P a 0) = constant_function (carrier R) a"
apply(rule ring_functions.function_ring_car_eqI[of R _ "carrier R"])
unfolding ring_functions_def apply(simp add: R.ring_axioms)
apply (simp add: assms to_fun_is_Fun)
using assms ring_functions.constant_function_closed[of R a "carrier R"]
unfolding ring_functions_def apply (simp add: R.ring_axioms)
using assms to_fun_const[of a ] unfolding constant_function_def
by auto
text‹Monomial polynomials map to monomial functions›
lemma monom_to_monomial:
assumes "a ∈ carrier R"
shows "to_fun (monom P a n) = monomial_function R a n"
apply(rule ring_functions.function_ring_car_eqI[of R _ "carrier R"])
unfolding ring_functions_def apply(simp add: R.ring_axioms)
apply (simp add: assms to_fun_is_Fun)
using assms U_function_ring.monomial_functions[of R a n] R.ring_axioms
unfolding U_function_ring_def
apply auto[1]
unfolding monomial_function_def
using assms to_fun_monom[of a _ n]
by auto
end
section‹Taylor Expansions›
subsection‹Monic Linear Polynomials›
text‹The polynomial representing the variable X›
definition X_poly where
"X_poly R = monom (UP R) 𝟭⇘R⇙ 1"
context UP_cring
begin
abbreviation(input) X where
"X ≡ X_poly R"
lemma X_closed:
"X ∈ carrier P"
unfolding X_poly_def
using P_def monom_closed by blast
lemma degree_X[simp]:
assumes "𝟭 ≠𝟬"
shows"degree X = 1"
unfolding X_poly_def
using assms P_def deg_monom[of 𝟭 1]
by blast
lemma X_not_zero:
assumes "𝟭 ≠𝟬"
shows"X ≠ 𝟬⇘P⇙"
using degree_X assms by force
lemma sub_X[simp]:
assumes "p ∈ carrier P"
shows "X of p = p"
unfolding X_poly_def
using P_def UP_pre_univ_prop.eval_monom1 assms compose_def to_poly_UP_pre_univ_prop
by metis
lemma sub_monom_deg_one:
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
shows "monom P a 1 of p = a ⊙⇘P⇙ p"
using assms sub_smult[of X p a] unfolding X_poly_def
by (metis P_def R.one_closed R.r_one X_closed X_poly_def monom_mult_smult sub_X)
lemma monom_rep_X_pow:
assumes "a ∈ carrier R"
shows "monom P a n = a⊙⇘P⇙(X[^]⇘P⇙n)"
proof-
have "monom P a n = a⊙⇘P⇙monom P 𝟭 n"
by (metis R.one_closed R.r_one assms monom_mult_smult)
then show ?thesis
unfolding X_poly_def
using monom_pow
by (simp add: P_def)
qed
lemma X_sub[simp]:
assumes "p ∈ carrier P"
shows "p of X = p"
apply(rule poly_induct3)
apply (simp add: assms)
using X_closed sub_add apply presburger
using sub_monom[of X] P_def monom_rep_X_pow X_closed by auto
text‹representation of monomials as scalar multiples of powers of X›
lemma ltrm_rep_X_pow:
assumes "p ∈ carrier P"
shows "ltrm p = (lcf p)⊙⇘P⇙(X[^]⇘P⇙(degree p))"
proof-
have "ltrm p = monom P (lcf p) (degree p)"
using assms unfolding leading_term_def by (simp add: P_def)
then show ?thesis
using monom_rep_X_pow P_def assms
by (simp add: cfs_closed)
qed
lemma to_fun_monom':
assumes "c ∈ carrier R"
assumes "c ≠𝟬"
assumes "x ∈ carrier R"
shows "to_fun (c ⊙⇘P⇙ X[^]⇘P⇙(n::nat)) x = c ⊗ x [^] n"
using P_def to_fun_monom monom_rep_X_pow UP_cring_axioms assms(1) assms(2) assms(3) by fastforce
lemma to_fun_X_pow:
assumes "x ∈ carrier R"
shows "to_fun (X[^]⇘P⇙(n::nat)) x = x [^] n"
using to_fun_monom'[of 𝟭 x n] assms
by (metis P.nat_pow_closed R.l_one R.nat_pow_closed R.one_closed R.r_null R.r_one
UP_one_closed X_closed to_fun_to_poly ring_hom_one smult_l_null smult_one to_poly_is_ring_hom)
end
text‹Monic linear polynomials›
definition X_poly_plus where
"X_poly_plus R a = (X_poly R) ⊕⇘(UP R)⇙ to_polynomial R a"
definition X_poly_minus where
"X_poly_minus R a = (X_poly R) ⊖⇘(UP R)⇙ to_polynomial R a"
context UP_cring
begin
abbreviation(input) X_plus where
"X_plus ≡ X_poly_plus R"
abbreviation(input) X_minus where
"X_minus ≡ X_poly_minus R"
lemma X_plus_closed:
assumes "a ∈ carrier R"
shows "(X_plus a) ∈ carrier P"
unfolding X_poly_plus_def using X_closed to_poly_closed
using P_def UP_a_closed assms by auto
lemma X_minus_closed:
assumes "a ∈ carrier R"
shows "(X_minus a) ∈ carrier P"
unfolding X_poly_minus_def using X_closed to_poly_closed
by (simp add: P_def UP_cring.UP_cring UP_cring_axioms assms cring.cring_simprules(4))
lemma X_minus_plus:
assumes "a ∈ carrier R"
shows "(X_minus a) = X_plus (⊖a)"
using P_def UP_ring.UP_ring UP_ring_axioms
by (simp add: X_poly_minus_def X_poly_plus_def a_minus_def assms to_poly_a_inv)
lemma degree_of_X_plus:
assumes "a ∈ carrier R"
assumes "𝟭 ≠𝟬"
shows "degree (X_plus a) = 1"
proof-
have 0:"degree (X_plus a) ≤ 1"
using deg_add degree_X P_def unfolding X_poly_plus_def
using UP_cring.to_poly_closed UP_cring_axioms X_closed assms(1) assms(2) by fastforce
have 1:"degree (X_plus a) > 0"
by (metis One_nat_def P_def R.one_closed R.r_zero X_poly_def
X_closed X_poly_plus_def X_plus_closed assms coeff_add coeff_monom deg_aboveD
gr0I lessI n_not_Suc_n to_polynomial_def to_poly_closed)
then show ?thesis
using "0" by linarith
qed
lemma degree_of_X_minus:
assumes "a ∈ carrier R"
assumes "𝟭 ≠𝟬"
shows "degree (X_minus a) = 1"
using degree_of_X_plus[of "⊖a"] X_minus_plus[simp] assms by auto
lemma ltrm_of_X:
shows"ltrm X = X"
unfolding leading_term_def
by (metis P_def R.one_closed X_poly_def is_UP_monom_def is_UP_monomI leading_term_def)
lemma ltrm_of_X_plus:
assumes "a ∈ carrier R"
assumes "𝟭 ≠𝟬"
shows "ltrm (X_plus a) = X"
unfolding X_poly_plus_def
using X_closed assms ltrm_of_sum_diff_degree[of X "to_poly a"]
degree_to_poly[of a] to_poly_closed[of a] degree_X ltrm_of_X
by (simp add: P_def)
lemma ltrm_of_X_minus:
assumes "a ∈ carrier R"
assumes "𝟭 ≠𝟬"
shows "ltrm (X_minus a) = X"
using X_minus_plus[of a] assms
by (simp add: ltrm_of_X_plus)
lemma lcf_of_X_minus:
assumes "a ∈ carrier R"
assumes "𝟭 ≠𝟬"
shows "lcf (X_minus a) = 𝟭"
using ltrm_of_X_minus unfolding X_poly_def
using P_def UP_cring.X_minus_closed UP_cring.lcf_eq UP_cring_axioms assms(1) assms(2) lcf_monom
by (metis R.one_closed)
lemma lcf_of_X_plus:
assumes "a ∈ carrier R"
assumes "𝟭 ≠𝟬"
shows "lcf (X_plus a) = 𝟭"
using ltrm_of_X_plus unfolding X_poly_def
by (metis lcf_of_X_minus P_def UP_cring.lcf_eq UP_cring.X_plus_closed UP_cring_axioms X_minus_closed assms(1) assms(2) degree_of_X_minus)
lemma to_fun_X[simp]:
assumes "a ∈ carrier R"
shows "to_fun X a = a"
using X_closed assms to_fun_sub_monom ltrm_is_UP_monom ltrm_of_X to_poly_closed
by (metis sub_X to_fun_to_poly)
lemma to_fun_X_plus[simp]:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "to_fun (X_plus a) b = b ⊕ a"
unfolding X_poly_plus_def
using assms to_fun_X[of b] to_fun_plus[of "to_poly a" X b] to_fun_to_poly[of a b]
using P_def X_closed to_poly_closed by auto
lemma to_fun_X_minus[simp]:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "to_fun (X_minus a) b = b ⊖ a"
using to_fun_X_plus[of "⊖ a" b] X_minus_plus[of a] assms
by (simp add: R.minus_eq)
lemma cfs_X_plus:
assumes "a ∈ carrier R"
shows "X_plus a n = (if n = 0 then a else (if n = 1 then 𝟭 else 𝟬))"
using assms cfs_add monom_closed UP_ring_axioms cfs_monom
unfolding X_poly_plus_def to_polynomial_def X_poly_def P_def
by auto
lemma cfs_X_minus:
assumes "a ∈ carrier R"
shows "X_minus a n = (if n = 0 then ⊖ a else (if n = 1 then 𝟭 else 𝟬))"
using cfs_X_plus[of "⊖ a"] assms
unfolding X_poly_plus_def X_poly_minus_def
by (simp add: P_def a_minus_def to_poly_a_inv)
text‹Linear substituions›
lemma X_plus_sub_deg:
assumes "a ∈ carrier R"
assumes "f ∈ carrier P"
shows "degree (f of (X_plus a)) = degree f"
apply(cases "𝟭 = 𝟬")
apply (metis P_def UP_one_closed X_plus_closed X_poly_def sub_X assms(1) assms(2) deg_one monom_one monom_zero sub_const)
using cring_sub_deg[of "X_plus a" f] assms X_plus_closed[of a] lcf_of_X_plus[of a]
ltrm_of_X_plus degree_of_X_plus[of a] P_def
by (metis lcf_eq R.nat_pow_one R.r_one UP_cring.cring_sub_deg UP_cring_axioms X_closed X_sub
cfs_closed coeff_simp deg_nzero_nzero degree_X lcoeff_nonzero2 sub_const)
lemma X_minus_sub_deg:
assumes "a ∈ carrier R"
assumes "f ∈ carrier P"
shows "degree (f of (X_minus a)) = degree f"
using X_plus_sub_deg[of "⊖a"] assms X_minus_plus[of a]
by simp
lemma plus_minus_sub:
assumes " a ∈ carrier R"
shows "X_plus a of X_minus a = X"
unfolding X_poly_plus_def
proof-
have "(X ⊕⇘P⇙ to_poly a) of X_minus a = (X of X_minus a) ⊕⇘P⇙ (to_poly a) of X_minus a"
using sub_add
by (simp add: X_closed X_minus_closed assms to_poly_closed)
then have "(X ⊕⇘P⇙ to_poly a) of X_minus a = (X_minus a) ⊕⇘P⇙ (to_poly a)"
by (simp add: X_minus_closed assms sub_to_poly)
then show "(X ⊕⇘UP R⇙ to_poly a) of X_minus a = X"
unfolding to_polynomial_def X_poly_minus_def
by (metis P.add.inv_solve_right P.minus_eq P_def
X_closed X_poly_minus_def X_minus_closed assms monom_closed to_polynomial_def)
qed
lemma minus_plus_sub:
assumes " a ∈ carrier R"
shows "X_minus a of X_plus a = X"
using plus_minus_sub[of "⊖a"]
unfolding X_poly_minus_def
unfolding X_poly_plus_def
using assms apply simp
by (metis P_def R.add.inv_closed R.minus_minus a_minus_def to_poly_a_inv)
lemma ltrm_times_X:
assumes "p ∈ carrier P"
shows "ltrm (X ⊗⇘P⇙ p) = X ⊗⇘P⇙ (ltrm p)"
using assms ltrm_of_X cring_ltrm_mult[of X p]
by (metis ltrm_deg_0 P.r_null R.l_one R.one_closed UP_cring.lcf_monom(1)
UP_cring_axioms X_closed X_poly_def cfs_closed deg_zero deg_ltrm monom_zero)
lemma times_X_not_zero:
assumes "p ∈ carrier P"
assumes "p ≠ 𝟬⇘P⇙"
shows "(X ⊗⇘P⇙ p) ≠ 𝟬⇘P⇙"
by (metis (no_types, opaque_lifting) lcf_monom(1) lcf_of_X_minus ltrm_of_X_minus P.inv_unique
P.r_null R.l_one R.one_closed UP_zero_closed X_closed zcf_def
zcf_zero_degree_zero assms(1) assms(2) cfs_closed cfs_zero cring_lcf_mult
deg_monom deg_nzero_nzero deg_ltrm degree_X degree_of_X_minus
monom_one monom_zero)
lemma degree_times_X:
assumes "p ∈ carrier P"
assumes "p ≠ 𝟬⇘P⇙"
shows "degree (X ⊗⇘P⇙ p) = degree p + 1"
using cring_deg_mult[of X p] assms times_X_not_zero[of p]
by (metis (no_types, lifting) P.r_null P.r_one P_def R.l_one R.one_closed
UP_cring.lcf_monom(1) UP_cring_axioms UP_zero_closed X_closed X_poly_def cfs_closed
deg_zero deg_ltrm degree_X monom_one monom_zero to_poly_inverse)
end
subsection‹Basic Facts About Taylor Expansions›
definition taylor_expansion where
"taylor_expansion R a p = compose R p (X_poly_plus R a)"
definition(in UP_cring) taylor where
"taylor ≡ taylor_expansion R"
context UP_cring
begin
lemma taylor_expansion_ring_hom:
assumes "c ∈ carrier R"
shows "taylor_expansion R c ∈ ring_hom P P"
unfolding taylor_expansion_def
using rev_sub_is_hom[of "X_plus c"]
unfolding rev_compose_def compose_def
using X_plus_closed assms by auto
notation taylor ("T⇘_⇙")
lemma(in UP_cring) taylor_closed:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "T⇘a⇙ f ∈ carrier P"
unfolding taylor_def
by (simp add: X_plus_closed assms(1) assms(2) sub_closed taylor_expansion_def)
lemma taylor_deg:
assumes "a ∈ carrier R"
assumes "p ∈ carrier P"
shows "degree (T⇘a⇙ p) = degree p"
unfolding taylor_def taylor_expansion_def
using X_plus_sub_deg[of a p] assms
by (simp add: taylor_expansion_def)
lemma taylor_id:
assumes "a ∈ carrier R"
assumes "p ∈ carrier P"
shows "p = (T⇘a⇙ p) of (X_minus a)"
unfolding taylor_expansion_def taylor_def
using assms sub_assoc[of p "X_plus a" "X_minus a"] X_plus_closed[of a] X_minus_closed[of a]
by (metis X_sub plus_minus_sub taylor_expansion_def)
lemma taylor_eval:
assumes "a ∈ carrier R"
assumes "f ∈ carrier P"
assumes "b ∈ carrier R"
shows "to_fun (T⇘a⇙ f) b = to_fun f (b ⊕ a)"
unfolding taylor_expansion_def taylor_def
using to_fun_sub[of "(X_plus a)" f b] to_fun_X_plus[of a b]
assms X_plus_closed[of a] by auto
lemma taylor_eval':
assumes "a ∈ carrier R"
assumes "f ∈ carrier P"
assumes "b ∈ carrier R"
shows "to_fun f (b) = to_fun (T⇘a⇙ f) (b ⊖ a) "
unfolding taylor_expansion_def taylor_def
using to_fun_sub[of "(X_minus a)" "T⇘a⇙ f" b] to_fun_X_minus[of b a]
assms X_minus_closed[of a]
by (metis taylor_closed taylor_def taylor_id taylor_expansion_def to_fun_X_minus)
lemma(in UP_cring) degree_monom:
assumes "a ∈ carrier R"
shows "degree (a ⊙⇘UP R⇙ (X_poly R)[^]⇘UP R⇙n) = (if a = 𝟬 then 0 else n)"
apply(cases "a = 𝟬")
apply (metis (full_types) P.nat_pow_closed P_def R.one_closed UP_smult_zero X_poly_def deg_zero monom_closed)
using P_def UP_cring.monom_rep_X_pow UP_cring_axioms assms deg_monom by fastforce
lemma(in UP_cring) poly_comp_finsum:
assumes "⋀i::nat. i ≤ n ⟹ g i ∈ carrier P"
assumes "q ∈ carrier P"
assumes "p = (⨁⇘P⇙ i ∈ {..n}. g i)"
shows "p of q = (⨁⇘P⇙ i ∈ {..n}. (g i) of q)"
proof-
have 0: "p of q = rev_sub q p"
unfolding compose_def rev_compose_def by blast
have 1: "p of q = finsum P (rev_compose R q ∘ g) {..n}"
unfolding 0 unfolding assms
apply(rule ring_hom_finsum[of "rev_compose R q" P "{..n}" g ])
using assms(2) rev_sub_is_hom apply blast
apply (simp add: UP_ring)
apply simp
by (simp add: assms(1))
show ?thesis unfolding 1
unfolding comp_apply rev_compose_def compose_def
by auto
qed
lemma(in UP_cring) poly_comp_expansion:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "degree p ≤ n"
shows "p of q = (⨁⇘P⇙ i ∈ {..n}. (p i) ⊙⇘P⇙ q[^]⇘P⇙i)"
proof-
obtain g where g_def: "g = (λi. monom P (p i) i)"
by blast
have 0: "⋀i. (g i) of q = (p i) ⊙⇘P⇙ q[^]⇘P⇙i"
proof- fix i show "g i of q = p i ⊙⇘P⇙ q [^]⇘P⇙ i"
using assms g_def P_def coeff_simp monom_sub
by (simp add: cfs_closed)
qed
have 1: "(⋀i. i ≤ n ⟹ g i ∈ carrier P)"
using g_def assms
by (simp add: cfs_closed)
have "(⨁⇘P⇙i∈{..n}. monom P (p i) i) = p"
using assms up_repr_le[of p n] coeff_simp[of p] unfolding P_def
by auto
then have "p = (⨁⇘P⇙ i ∈ {..n}. g i)"
using g_def by auto
then have "p of q = (⨁⇘P⇙i∈{..n}. g i of q)"
using 0 1 poly_comp_finsum[of n g q p]
using assms(2)
by blast
then show ?thesis
by(simp add: 0)
qed
lemma(in UP_cring) taylor_sum:
assumes "p ∈ carrier P"
assumes "degree p ≤ n"
assumes "a ∈ carrier R"
shows "p = (⨁⇘P⇙ i ∈ {..n}. T⇘a⇙ p i ⊙⇘P⇙ (X_minus a)[^]⇘P⇙i)"
proof-
have 0: "(T⇘a⇙ p) of X_minus a = p"
using P_def taylor_id assms(1) assms(3)
by fastforce
have 1: "degree (T⇘a⇙ p) ≤ n"
using assms
by (simp add: taylor_deg)
have 2: "T⇘a⇙ p of X_minus a = (⨁⇘P⇙i∈{..n}. T⇘a⇙ p i ⊙⇘P⇙ X_minus a [^]⇘P⇙ i)"
using 1 X_minus_closed[of a] poly_comp_expansion[of "T⇘a⇙ p" "X_minus a" n]
assms taylor_closed
by blast
then show ?thesis
using 0
by simp
qed
text‹The $i^{th}$ term in the taylor expansion›
definition taylor_term where
"taylor_term c p i = (taylor_expansion R c p i) ⊙⇘UP R⇙ (UP_cring.X_minus R c) [^]⇘UP R⇙i"
lemma (in UP_cring) taylor_term_closed:
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
shows "taylor_term a p i ∈ carrier (UP R)"
unfolding taylor_term_def
using P.nat_pow_closed P_def taylor_closed taylor_def X_minus_closed assms(1) assms(2) smult_closed
by (simp add: cfs_closed)
lemma(in UP_cring) taylor_term_sum:
assumes "p ∈ carrier P"
assumes "degree p ≤ n"
assumes "a ∈ carrier R"
shows "p = (⨁⇘P⇙ i ∈ {..n}. taylor_term a p i)"
unfolding taylor_term_def taylor_def
using assms taylor_sum[of p n a] P_def
using taylor_def by auto
lemma (in UP_cring) taylor_expansion_add:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "c ∈ carrier R"
shows "taylor_expansion R c (p ⊕⇘UP R⇙ q) = (taylor_expansion R c p) ⊕⇘UP R⇙ (taylor_expansion R c q)"
unfolding taylor_expansion_def
using assms X_plus_closed[of c] P_def sub_add
by blast
lemma (in UP_cring) taylor_term_add:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "a ∈ carrier R"
shows "taylor_term a (p ⊕⇘UP R⇙q) i = taylor_term a p i ⊕⇘UP R⇙ taylor_term a q i"
using assms taylor_expansion_add[of p q a]
unfolding taylor_term_def
using P.nat_pow_closed P_def taylor_closed X_minus_closed cfs_add smult_l_distr
by (simp add: taylor_def cfs_closed)
lemma (in UP_cring) to_fun_taylor_term:
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
assumes "c ∈ carrier R"
shows "to_fun (taylor_term c p i) a = (T⇘c⇙ p i) ⊗ (a ⊖ c)[^]i"
using assms to_fun_smult[of "X_minus c [^]⇘UP R⇙ i" a "taylor_expansion R c p i"]
to_fun_X_minus[of c a] to_fun_nat_pow[of "X_minus c" a i]
unfolding taylor_term_def
using P.nat_pow_closed P_def taylor_closed taylor_def X_minus_closed
by (simp add: cfs_closed)
end
subsection‹Defining the (Scalar-Valued) Derivative of a Polynomial Using the Taylor Expansion›
definition derivative where
"derivative R f a = (taylor_expansion R a f) 1"
context UP_cring
begin
abbreviation(in UP_cring) deriv where
"deriv ≡ derivative R"
lemma(in UP_cring) deriv_closed:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "(derivative R f a) ∈ carrier R"
unfolding derivative_def
using taylor_closed taylor_def assms(1) assms(2) cfs_closed by auto
lemma(in UP_cring) deriv_add:
assumes "f ∈ carrier P"
assumes "g ∈ carrier P"
assumes "a ∈ carrier R"
shows "deriv (f ⊕⇘P⇙ g) a = deriv f a ⊕ deriv g a"
unfolding derivative_def taylor_expansion_def using assms
by (simp add: X_plus_closed sub_add sub_closed)
end
section‹The Polynomial-Valued Derivative Operator›
context UP_cring
begin
subsection‹Operator Which Shifts Coefficients›
lemma cfs_times_X:
assumes "g ∈ carrier P"
shows "(X ⊗⇘P⇙ g) (Suc n) = g n"
apply(rule poly_induct3[of g])
apply (simp add: assms)
apply (metis (no_types, lifting) P.m_closed P.r_distr X_closed cfs_add)
by (metis (no_types, lifting) P_def R.l_one R.one_closed R.r_null Suc_eq_plus1 X_poly_def
cfs_monom coeff_monom_mult coeff_simp monom_closed monom_mult)
lemma times_X_pow_coeff:
assumes "g ∈ carrier P"
shows "(monom P 𝟭 k ⊗⇘P⇙ g) (n + k) = g n"
using coeff_monom_mult P.m_closed P_def assms coeff_simp monom_closed
by (simp add: cfs_closed)
lemma zcf_eq_zero_unique:
assumes "f ∈ carrier P"
assumes "g ∈ carrier P ∧ (f = X ⊗⇘P⇙ g)"
shows "⋀ h. h ∈ carrier P ∧ (f = X ⊗⇘P⇙ h) ⟹ h = g"
proof-
fix h
assume A: "h ∈ carrier P ∧ (f = X ⊗⇘P⇙ h)"
then have 0: " X ⊗⇘P⇙ g = X ⊗⇘P⇙ h"
using assms(2) by auto
show "h = g"
using 0 A assms
by (metis P_def coeff_simp cfs_times_X up_eqI)
qed
lemma f_minus_ctrm:
assumes "f ∈ carrier P"
shows "zcf(f ⊖⇘P⇙ ctrm f) = 𝟬"
using assms
by (smt (verit) ctrm_is_poly P.add.inv_closed P.minus_closed P_def R.r_neg R.zero_closed zcf_to_fun
to_fun_minus to_fun_plus UP_cring_axioms zcf_ctrm zcf_def a_minus_def cfs_closed)
definition poly_shift where
"poly_shift f n = f (Suc n)"
lemma poly_shift_closed:
assumes "f ∈ carrier P"
shows "poly_shift f ∈ carrier P"
apply(rule UP_car_memI[of "deg R f"])
unfolding poly_shift_def
proof -
fix n :: nat
assume "deg R f < n"
then have "deg R f < Suc n"
using Suc_lessD by blast
then have "f (Suc n) = 𝟬⇘P⇙ (Suc n)"
by (metis P.l_zero UP_zero_closed assms coeff_of_sum_diff_degree0)
then show "f (Suc n) = 𝟬"
by simp
next
show " ⋀n. f (Suc n) ∈ carrier R"
by(rule cfs_closed, rule assms)
qed
lemma poly_shift_eq_0:
assumes "f ∈ carrier P"
shows "f n = (ctrm f ⊕⇘P⇙ X ⊗⇘P⇙ poly_shift f) n"
apply(cases "n = 0")
apply (smt (verit) ctrm_degree ctrm_is_poly ltrm_of_X One_nat_def P.r_null P.r_zero P_def UP_cring.lcf_monom(1) UP_cring_axioms UP_mult_closed UP_r_one UP_zero_closed X_closed zcf_ltrm_mult zcf_def zcf_zero assms cfs_add cfs_closed deg_zero degree_X lessI monom_one poly_shift_closed to_poly_inverse)
proof- assume A: "n ≠ 0"
then obtain k where k_def: " n = Suc k"
by (meson lessI less_Suc_eq_0_disj)
show ?thesis
using cfs_times_X[of "poly_shift f" k] poly_shift_def[of f k] poly_shift_closed assms
cfs_add[of "ctrm f" "X ⊗⇘P⇙ poly_shift f" n] unfolding k_def
by (simp add: X_closed cfs_closed cfs_monom)
qed
lemma poly_shift_eq:
assumes "f ∈ carrier P"
shows "f = (ctrm f ⊕⇘P⇙ X ⊗⇘P⇙ poly_shift f)"
by(rule ext, rule poly_shift_eq_0, rule assms)
lemma poly_shift_id:
assumes "f ∈ carrier P"
shows "f ⊖⇘P⇙ ctrm f = X ⊗⇘P⇙ poly_shift f"
using assms poly_shift_eq[of f] poly_shift_closed unfolding a_minus_def
by (metis ctrm_is_poly P.add.inv_solve_left P.m_closed UP_a_comm UP_a_inv_closed X_closed)
lemma poly_shift_degree_zero:
assumes "p ∈ carrier P"
assumes "degree p = 0"
shows "poly_shift p = 𝟬⇘P⇙"
by (metis ltrm_deg_0 P.r_neg P.r_null UP_ring UP_zero_closed X_closed zcf_eq_zero_unique
abelian_group.minus_eq assms(1) assms(2) poly_shift_closed poly_shift_id ring_def)
lemma poly_shift_degree:
assumes "p ∈ carrier P"
assumes "degree p > 0"
shows "degree (poly_shift p) = degree p - 1 "
using poly_shift_id[of p]
by (metis ctrm_degree ctrm_is_poly P.r_null X_closed add_diff_cancel_right' assms(1) assms(2)
deg_zero degree_of_difference_diff_degree degree_times_X nat_less_le poly_shift_closed)
lemma poly_shift_monom:
assumes "a ∈ carrier R"
shows "poly_shift (monom P a (Suc k)) = (monom P a k)"
proof-
have "(monom P a (Suc k)) = ctrm (monom P a (Suc k)) ⊕⇘P⇙ X ⊗⇘P⇙poly_shift (monom P a (Suc k))"
using poly_shift_eq[of "monom P a (Suc k)"] assms monom_closed
by blast
then have "(monom P a (Suc k)) = 𝟬⇘P⇙ ⊕⇘P⇙ X ⊗⇘P⇙poly_shift (monom P a (Suc k))"
using assms by simp
then have "(monom P a (Suc k)) = X ⊗⇘P⇙poly_shift (monom P a (Suc k))"
using X_closed assms poly_shift_closed by auto
then have "X ⊗⇘P⇙(monom P a k) = X ⊗⇘P⇙poly_shift (monom P a (Suc k))"
by (metis P_def R.l_one R.one_closed X_poly_def assms monom_mult plus_1_eq_Suc)
then show ?thesis
using X_closed X_not_zero assms
by (meson UP_mult_closed zcf_eq_zero_unique monom_closed poly_shift_closed)
qed
lemma(in UP_cring) poly_shift_add:
assumes "f ∈ carrier P"
assumes "g ∈ carrier P"
shows "poly_shift (f ⊕⇘P⇙ g) = (poly_shift f) ⊕⇘P⇙ (poly_shift g)"
apply(rule ext)
using cfs_add[of "poly_shift f" "poly_shift g"] poly_shift_closed poly_shift_def
by (simp add: poly_shift_def assms(1) assms(2))
lemma(in UP_cring) poly_shift_s_mult:
assumes "f ∈ carrier P"
assumes "s ∈ carrier R"
shows "poly_shift (s ⊙⇘P⇙f) = s ⊙⇘P⇙ (poly_shift f)"
proof-
have "(s ⊙⇘P⇙f) = (ctrm (s ⊙⇘P⇙f)) ⊕⇘P⇙(X ⊗⇘P⇙ poly_shift (s ⊙⇘P⇙f))"
using poly_shift_eq[of "(s ⊙⇘P⇙f)"] assms(1) assms(2)
by blast
then have 0: "(s ⊙⇘P⇙f) = (s ⊙⇘P⇙(ctrm f)) ⊕⇘P⇙(X ⊗⇘P⇙ poly_shift (s ⊙⇘P⇙f))"
using ctrm_smult assms(1) assms(2) by auto
have 1: "(s ⊙⇘P⇙f) = s ⊙⇘P⇙ ((ctrm f) ⊕⇘P⇙ (X ⊗⇘P⇙ (poly_shift f)))"
using assms(1) poly_shift_eq by auto
have 2: "(s ⊙⇘P⇙f) = (s ⊙⇘P⇙(ctrm f)) ⊕⇘P⇙ (s ⊙⇘P⇙(X ⊗⇘P⇙ (poly_shift f)))"
by (simp add: "1" X_closed assms(1) assms(2) ctrm_is_poly poly_shift_closed smult_r_distr)
have 3: "(s ⊙⇘P⇙f) = (s ⊙⇘P⇙(ctrm f)) ⊕⇘P⇙ (X ⊗⇘P⇙ (s ⊙⇘P⇙(poly_shift f)))"
using "2" UP_m_comm X_closed assms(1) assms(2) smult_assoc2
by (simp add: poly_shift_closed)
have 4: "(X ⊗⇘P⇙ poly_shift (s ⊙⇘P⇙f)) = (X ⊗⇘P⇙ (s ⊙⇘P⇙(poly_shift f)))"
using 3 0 X_closed assms(1) assms(2) ctrm_is_poly poly_shift_closed
by auto
then show ?thesis
using X_closed X_not_zero assms(1) assms(2)
by (metis UP_mult_closed UP_smult_closed zcf_eq_zero_unique poly_shift_closed)
qed
lemma zcf_poly_shift:
assumes "f ∈ carrier P"
shows "zcf (poly_shift f) = f 1"
apply(rule poly_induct3)
apply (simp add: assms)
using poly_shift_add zcf_add cfs_add poly_shift_closed apply metis
unfolding zcf_def using poly_shift_monom poly_shift_degree_zero
by (simp add: poly_shift_def)
fun poly_shift_iter ("shift") where
Base:"poly_shift_iter 0 f = f"|
Step:"poly_shift_iter (Suc n) f = poly_shift (poly_shift_iter n f)"
lemma shift_closed:
assumes "f ∈ carrier P"
shows "shift n f ∈ carrier P"
apply(induction n)
using assms poly_shift_closed by auto
subsection‹Operator Which Multiplies Coefficients by Their Degree›
definition n_mult where
"n_mult f = (λn. [n]⋅⇘R⇙(f n))"
lemma(in UP_cring) n_mult_closed:
assumes "f ∈ carrier P"
shows "n_mult f ∈ carrier P"
apply(rule UP_car_memI[of "deg R f"])
unfolding n_mult_def
apply (metis P.l_zero R.add.nat_pow_one UP_zero_closed assms cfs_zero coeff_of_sum_diff_degree0)
using assms cfs_closed by auto
text‹Facts about the shift function›
lemma shift_one:
"shift (Suc 0) = poly_shift"
by auto
lemma shift_factor0:
assumes "f ∈ carrier P"
shows "degree f ≥ (Suc k) ⟹ degree (f ⊖⇘P⇙ ((shift (Suc k) f) ⊗⇘P⇙(X[^]⇘P⇙(Suc k)))) < (Suc k)"
proof(induction k)
case 0
have 0: " f ⊖⇘P⇙ (ctrm f) = (shift (Suc 0) f)⊗⇘P⇙X"
by (metis UP_m_comm X_closed assms poly_shift_id shift_closed shift_one)
then have " f ⊖⇘P⇙(shift (Suc 0) f)⊗⇘P⇙X = (ctrm f) "
proof-
have " f ⊖⇘P⇙ (ctrm f) ⊖⇘P⇙ (shift (Suc 0) f)⊗⇘P⇙X= (shift (Suc 0) f)⊗⇘P⇙X ⊖⇘P⇙ (shift (Suc 0) f)⊗⇘P⇙X"
using 0 by simp
then have " f ⊖⇘P⇙ (ctrm f) ⊖⇘P⇙ (shift (Suc 0) f)⊗⇘P⇙X = 𝟬⇘P⇙"
using UP_cring.UP_cring[of R] assms
by (metis "0" P.ring_simprules(4) P_def UP_ring.UP_ring UP_ring_axioms
a_minus_def abelian_group.r_neg ctrm_is_poly ring_def)
then have " f ⊖⇘P⇙ ((ctrm f) ⊕⇘P⇙ (shift (Suc 0) f)⊗⇘P⇙X) = 𝟬⇘P⇙"
using assms P.ring_simprules
by (metis "0" poly_shift_id poly_shift_eq)
then have " f ⊖⇘P⇙ ((shift (Suc 0) f)⊗⇘P⇙X ⊕⇘P⇙ (ctrm f) ) = 𝟬⇘P⇙"
using P.m_closed UP_a_comm X_closed assms ctrm_is_poly shift_closed
by presburger
then have "f ⊖⇘P⇙ ((shift (Suc 0) f)⊗⇘P⇙X) ⊖⇘P⇙ (ctrm f)= 𝟬⇘P⇙"
using P.add.m_assoc P.ring_simprules(14) P.ring_simprules(19) assms "0"
P.add.inv_closed P.r_neg P.r_zero ctrm_is_poly
by (smt (verit, ccfv_threshold))
then show ?thesis
by (metis "0" P.add.m_comm P.m_closed P.ring_simprules(14) P.ring_simprules(18)
P.ring_simprules(3) X_closed assms ctrm_is_poly poly_shift_id poly_shift_eq
shift_closed)
qed
then have " f ⊖⇘P⇙(shift (Suc 0) f)⊗⇘P⇙(X[^]⇘P⇙(Suc 0)) = (ctrm f) "
proof-
have "X = X[^]⇘P⇙(Suc 0)"
by (simp add: X_closed)
then show ?thesis
using 0 ‹f ⊖⇘P⇙ shift (Suc 0) f ⊗⇘P⇙ X = ctrm f›
by auto
qed
then have " degree (f ⊖⇘P⇙(shift (Suc 0) f)⊗⇘P⇙(X[^]⇘P⇙(Suc 0))) < 1"
using ctrm_degree[of f] assms by simp
then show ?case
by blast
next
case (Suc n)
fix k
assume IH: "degree f ≥ (Suc k) ⟹ degree (f ⊖⇘P⇙ ((shift (Suc k) f) ⊗⇘P⇙(X[^]⇘P⇙(Suc k)))) < (Suc k)"
show "degree f ≥ (Suc (Suc k)) ⟹ degree (f ⊖⇘P⇙ ((shift (Suc (Suc k)) f) ⊗⇘P⇙(X[^]⇘P⇙(Suc (Suc k))))) < (Suc (Suc k))"
proof-
obtain n where n_def: "n = Suc k"
by simp
have IH': "degree f ≥ n ⟹ degree (f ⊖⇘P⇙ ((shift n f) ⊗⇘P⇙(X[^]⇘P⇙n))) < n"
using n_def IH by auto
have P: "degree f ≥ (Suc n) ⟹ degree (f ⊖⇘P⇙ ((shift (Suc n) f) ⊗⇘P⇙(X[^]⇘P⇙(Suc n)))) < (Suc n)"
proof-
obtain g where g_def: "g = (f ⊖⇘P⇙ ((shift n f) ⊗⇘P⇙(X[^]⇘P⇙n)))"
by simp
obtain s where s_def: "s = shift n f"
by simp
obtain s' where s'_def: "s' = shift (Suc n) f"
by simp
have P: "g ∈ carrier P" "s ∈ carrier P" "s' ∈ carrier P" "(X[^]⇘P⇙n) ∈ carrier P"
using s_def s'_def g_def assms shift_closed[of f n]
apply (simp add: X_closed)
apply (simp add: ‹f ∈ carrier P ⟹ shift n f ∈ carrier P› assms s_def)
using P_def UP_cring.shift_closed UP_cring_axioms assms s'_def apply blast
using X_closed by blast
have g_def': "g = (f ⊖⇘P⇙ (s ⊗⇘P⇙(X[^]⇘P⇙n)))"
using g_def s_def by auto
assume "degree f ≥ (Suc n)"
then have " degree (f ⊖⇘P⇙ (s ⊗⇘P⇙(X[^]⇘P⇙n))) < n"
using IH' Suc_leD s_def by blast
then have d_g: "degree g < n" using g_def' by auto
have P0: "f ⊖⇘P⇙ (s' ⊗⇘P⇙(X[^]⇘P⇙(Suc n))) = ((ctrm s)⊗⇘P⇙(X[^]⇘P⇙n)) ⊕⇘P⇙ g"
proof-
have "s = (ctrm s) ⊕⇘P⇙ (X ⊗⇘P⇙ s')"
using s_def s'_def P_def poly_shift_eq UP_cring_axioms assms shift_closed
by (simp add: UP_cring.poly_shift_eq)
then have 0: "g = f ⊖⇘P⇙ ((ctrm s) ⊕⇘P⇙ (X ⊗⇘P⇙ s')) ⊗⇘P⇙(X[^]⇘P⇙n)"
using g_def' by auto
then have "g = f ⊖⇘P⇙ ((ctrm s)⊗⇘P⇙(X[^]⇘P⇙n)) ⊖⇘P⇙ ((X ⊗⇘P⇙ s') ⊗⇘P⇙(X[^]⇘P⇙n))"
using P cring_axioms X_closed P.l_distr P.ring_simprules(19) UP_a_assoc a_minus_def assms
by (simp add: a_minus_def ctrm_is_poly)
then have "g ⊕⇘P⇙ ((X ⊗⇘P⇙ s') ⊗⇘P⇙(X[^]⇘P⇙n)) = f ⊖⇘P⇙ ((ctrm s)⊗⇘P⇙(X[^]⇘P⇙n))"
using P cring_axioms X_closed P.l_distr P.ring_simprules UP_a_assoc a_minus_def assms
by (simp add: P.r_neg2 ctrm_is_poly)
then have " ((ctrm s)⊗⇘P⇙(X[^]⇘P⇙n)) = f ⊖⇘P⇙ (g ⊕⇘P⇙ ((X ⊗⇘P⇙ s') ⊗⇘P⇙(X[^]⇘P⇙n)))"
using P cring_axioms X_closed P.ring_simprules UP_a_assoc a_minus_def assms
by (simp add: P.ring_simprules(17) ctrm_is_poly)
then have " ((ctrm s)⊗⇘P⇙(X[^]⇘P⇙n)) = f ⊖⇘P⇙ (((X ⊗⇘P⇙ s') ⊗⇘P⇙(X[^]⇘P⇙n)) ⊕⇘P⇙ g)"
by (simp add: P(1) P(3) UP_a_comm X_closed)
then have "((ctrm s)⊗⇘P⇙(X[^]⇘P⇙n)) = f ⊖⇘P⇙ ((X ⊗⇘P⇙ s') ⊗⇘P⇙(X[^]⇘P⇙n)) ⊖⇘P⇙ g"
using P(1) P(3) P.ring_simprules(19) UP_a_assoc a_minus_def assms
by (simp add: a_minus_def X_closed)
then have "((ctrm s)⊗⇘P⇙(X[^]⇘P⇙n)) ⊕⇘P⇙ g= f ⊖⇘P⇙ ((X ⊗⇘P⇙ s') ⊗⇘P⇙(X[^]⇘P⇙n))"
by (metis P(1) P(3) P(4) P.add.inv_solve_right P.m_closed P.ring_simprules(14)
P.ring_simprules(4) P_def UP_cring.X_closed UP_cring_axioms assms)
then have "((ctrm s)⊗⇘P⇙(X[^]⇘P⇙n)) ⊕⇘P⇙ g= f ⊖⇘P⇙ ((s' ⊗⇘P⇙ X) ⊗⇘P⇙(X[^]⇘P⇙n))"
by (simp add: P(3) UP_m_comm X_closed)
then have "((ctrm s)⊗⇘P⇙(X[^]⇘P⇙n)) ⊕⇘P⇙ g= f ⊖⇘P⇙ (s' ⊗⇘P⇙(X[^]⇘P⇙(Suc n)))"
using P(3) P.nat_pow_Suc2 UP_m_assoc X_closed by auto
then show ?thesis
by auto
qed
have P1: "degree (((ctrm s)⊗⇘P⇙(X[^]⇘P⇙n)) ⊕⇘P⇙ g) ≤ n"
proof-
have Q0: "degree ((ctrm s)⊗⇘P⇙(X[^]⇘P⇙n)) ≤ n"
proof(cases "ctrm s = 𝟬⇘P⇙")
case True
then show ?thesis
by (simp add: P(4))
next
case False
then have F0: "degree ((ctrm s)⊗⇘P⇙(X[^]⇘P⇙n)) ≤ degree (ctrm s) + degree (X[^]⇘P⇙n) "
by (meson ctrm_is_poly P(2) P(4) deg_mult_ring)
have F1: "𝟭≠𝟬⟹ degree (X[^]⇘P⇙n) = n"
unfolding X_poly_def
using P_def cring_monom_degree by auto
show ?thesis
by (metis (no_types, opaque_lifting) F0 F1 ltrm_deg_0 P(2) P.r_null P_def R.l_null R.l_one
R.nat_pow_closed R.zero_closed X_poly_def assms cfs_closed
add_0 deg_const deg_zero deg_ltrm
monom_pow monom_zero zero_le)
qed
then show ?thesis
using d_g
by (simp add: P(1) P(2) P(4) bound_deg_sum ctrm_is_poly)
qed
then show ?thesis
using s'_def P0 by auto
qed
assume "degree f ≥ (Suc (Suc k)) "
then show "degree (f ⊖⇘P⇙ ((shift (Suc (Suc k)) f) ⊗⇘P⇙(X[^]⇘P⇙(Suc (Suc k))))) < (Suc (Suc k))"
using P by(simp add: n_def)
qed
qed
lemma(in UP_cring) shift_degree0:
assumes "f ∈ carrier P"
shows "degree f >n ⟹ Suc (degree (shift (Suc n) f)) = degree (shift n f)"
proof(induction n)
case 0
assume B: "0< degree f"
have 0: "degree (shift 0 f) = degree f"
by simp
have 1: "degree f = degree (f ⊖⇘P⇙ (ctrm f))"
using assms(1) B ctrm_degree degree_of_difference_diff_degree
by (simp add: ctrm_is_poly)
have "(f ⊖⇘P⇙ (ctrm f)) = X ⊗⇘P⇙(shift 1 f)"
using P_def poly_shift_id UP_cring_axioms assms(1) by auto
then have "degree (f ⊖⇘P⇙ (ctrm f)) = 1 + (degree (shift 1 f))"
by (metis "1" B P.r_null X_closed add.commute assms deg_nzero_nzero degree_times_X not_gr_zero shift_closed)
then have "degree (shift 0 f) = 1 + (degree (shift 1 f))"
using 0 1 by auto
then show ?case
by simp
next
case (Suc n)
fix n
assume IH: "(n < degree f ⟹ Suc (degree (shift (Suc n) f)) = degree (shift n f))"
show "Suc n < degree f ⟹ Suc (degree (shift (Suc (Suc n)) f)) = degree (shift (Suc n) f)"
proof-
assume A: " Suc n < degree f"
then have 0: "(shift (Suc n) f) = ctrm ((shift (Suc n) f)) ⊕⇘P⇙ (shift (Suc (Suc n)) f)⊗⇘P⇙X"
by (metis UP_m_comm X_closed assms local.Step poly_shift_eq shift_closed)
have N: "(shift (Suc (Suc n)) f) ≠ 𝟬⇘P⇙"
proof
assume C: "shift (Suc (Suc n)) f = 𝟬⇘P⇙"
obtain g where g_def: "g = f ⊖⇘P⇙ (shift (Suc (Suc n)) f)⊗⇘P⇙(X[^]⇘P⇙(Suc (Suc n)))"
by simp
have C0: "degree g < degree f"
using g_def assms A
by (meson Suc_leI Suc_less_SucD Suc_mono less_trans_Suc shift_factor0)
have C1: "g = f"
using C
by (simp add: P.minus_eq X_closed assms g_def)
then show False
using C0 by auto
qed
have 1: "degree (shift (Suc n) f) = degree ((shift (Suc n) f) ⊖⇘P⇙ ctrm ((shift (Suc n) f)))"
proof(cases "degree (shift (Suc n) f) = 0")
case True
then show ?thesis
using N assms poly_shift_degree_zero poly_shift_closed shift_closed by auto
next
case False
then have "degree (shift (Suc n) f) > degree (ctrm ((shift (Suc n) f)))"
proof -
have "shift (Suc n) f ∈ carrier P"
using assms shift_closed by blast
then show ?thesis
using False ctrm_degree by auto
qed
then show ?thesis
proof -
show ?thesis
using ‹degree (ctrm (shift (Suc n) f)) < degree (shift (Suc n) f)›
assms ctrm_is_poly degree_of_difference_diff_degree shift_closed by presburger
qed
qed
have 2: "(shift (Suc n) f) ⊖⇘P⇙ ctrm ((shift (Suc n) f)) = (shift (Suc (Suc n)) f)⊗⇘P⇙X"
using 0
by (metis Cring_Poly.INTEG.Step P.m_comm X_closed assms poly_shift_id shift_closed)
have 3: "degree ((shift (Suc n) f) ⊖⇘P⇙ ctrm ((shift (Suc n) f))) = degree (shift (Suc (Suc n)) f) + 1"
using 2 N X_closed X_not_zero assms degree_X shift_closed
by (metis UP_m_comm degree_times_X)
then show ?thesis using 1
by linarith
qed
qed
lemma(in UP_cring) shift_degree:
assumes "f ∈ carrier P"
shows "degree f ≥ n ⟹ degree (shift n f) + n = degree f"
proof(induction n)
case 0
then show ?case
by auto
next
case (Suc n)
fix n
assume IH: "(n ≤ degree f ⟹ degree (shift n f) + n = degree f)"
show "Suc n ≤ degree f ⟹ degree (shift (Suc n) f) + Suc n = degree f"
proof-
assume A: "Suc n ≤ degree f "
have 0: "degree (shift n f) + n = degree f"
using IH A by auto
have 1: "degree (shift n f) = Suc (degree (shift (Suc n) f))"
using A assms shift_degree0 by auto
show "degree (shift (Suc n) f) + Suc n = degree f"
using 0 1 by simp
qed
qed
lemma(in UP_cring) shift_degree':
assumes "f ∈ carrier P"
shows "degree (shift (degree f) f) = 0"
using shift_degree assms
by fastforce
lemma(in UP_cring) shift_above_degree:
assumes "f ∈ carrier P"
assumes "k > degree f"
shows "(shift k f) = 𝟬⇘P⇙"
proof-
have "⋀n. shift ((degree f)+ (Suc n)) f = 𝟬⇘P⇙"
proof-
fix n
show "shift ((degree f)+ (Suc n)) f = 𝟬⇘P⇙"
proof(induction n)
case 0
have B0:"shift (degree f) f = ctrm(shift (degree f) f) ⊕⇘P⇙ (shift (degree f + Suc 0) f)⊗⇘P⇙X"
proof -
have f1: "∀f n. f ∉ carrier P ∨ shift n f ∈ carrier P"
by (meson shift_closed)
then have "shift (degree f + Suc 0) f ∈ carrier P"
using assms(1) by blast
then show ?thesis
using f1 by (simp add: P.m_comm X_closed assms(1) poly_shift_eq)
qed
have B1:"shift (degree f) f = ctrm(shift (degree f) f)"
proof -
have "shift (degree f) f ∈ carrier P"
using assms(1) shift_closed by blast
then show ?thesis
using ltrm_deg_0 assms(1) shift_degree' by auto
qed
have B2: "(shift (degree f + Suc 0) f)⊗⇘P⇙X = 𝟬⇘P⇙"
using B0 B1 X_closed assms(1)
proof -
have "∀f n. f ∉ carrier P ∨ shift n f ∈ carrier P"
using shift_closed by blast
then show ?thesis
by (metis (no_types) B0 B1 P.add.l_cancel_one UP_mult_closed X_closed assms(1))
qed
then show ?case
by (metis P.r_null UP_m_comm UP_zero_closed X_closed assms(1) zcf_eq_zero_unique shift_closed)
next
case (Suc n)
fix n
assume "shift (degree f + Suc n) f = 𝟬⇘P⇙"
then show "shift (degree f + Suc (Suc n)) f = 𝟬⇘P⇙"
by (simp add: poly_shift_degree_zero)
qed
qed
then show ?thesis
using assms(2) less_iff_Suc_add by auto
qed
lemma(in UP_domain) shift_cfs0:
assumes "f ∈ carrier P"
shows "zcf(shift 1 f) = f 1"
using assms
by (simp add: zcf_poly_shift)
lemma(in UP_cring) X_mult_cf:
assumes "p ∈ carrier P"
shows "(p ⊗⇘P⇙ X) (k+1) = p k"
unfolding X_poly_def
using assms
by (metis UP_m_comm X_closed X_poly_def add.commute plus_1_eq_Suc cfs_times_X)
lemma(in UP_cring) X_pow_cf:
assumes "p ∈ carrier P"
shows "(p ⊗⇘P⇙ X[^]⇘P⇙(n::nat)) (n + k) = p k"
proof-
have P: "⋀f. f ∈ carrier P ⟹ (f ⊗⇘P⇙ X[^]⇘P⇙(n::nat)) (n + k) = f k"
proof(induction n)
show "⋀f. f ∈ carrier P ⟹ (f ⊗⇘P⇙ X [^]⇘P⇙ (0::nat)) (0 + k) = f k"
proof-
fix f
assume B0: "f ∈ carrier P"
show "(f ⊗⇘P⇙ X [^]⇘P⇙ (0::nat)) (0 + k) = f k"
by (simp add: B0)
qed
fix n
fix f
assume IH: "(⋀f. f ∈ carrier P ⟹ (f ⊗⇘P⇙ X [^]⇘P⇙ n) (n + k) = f k)"
assume A0: " f ∈ carrier P"
show "(f ⊗⇘P⇙ X [^]⇘P⇙ Suc n) (Suc n + k) = f k"
proof-
have 0: "(f ⊗⇘P⇙ X [^]⇘P⇙ n)(n + k) = f k"
using A0 IH by simp
have 1: "((f ⊗⇘P⇙ X [^]⇘P⇙ n)⊗⇘P⇙X) (Suc n + k) = (f ⊗⇘P⇙ X [^]⇘P⇙ n)(n + k)"
using X_mult_cf A0 P.m_closed P.nat_pow_closed
Suc_eq_plus1 X_closed add_Suc by presburger
have 2: "(f ⊗⇘P⇙ (X [^]⇘P⇙ n ⊗⇘P⇙X)) (Suc n + k) = (f ⊗⇘P⇙ X [^]⇘P⇙ n)(n + k)"
using 1
by (simp add: A0 UP_m_assoc X_closed)
then show ?thesis
by (simp add: "0")
qed
qed
show ?thesis using assms P[of p] by auto
qed
lemma poly_shift_cfs:
assumes "f ∈ carrier P"
shows "poly_shift f n = f (Suc n)"
proof-
have "(f ⊖⇘P⇙ ctrm f) (Suc n) = (X ⊗⇘P⇙ (poly_shift f)) (Suc n)"
using assms poly_shift_id by auto
then show ?thesis unfolding X_poly_def using poly_shift_closed assms
by (metis (no_types, lifting) ctrm_degree ctrm_is_poly
P.add.m_comm P.minus_closed coeff_of_sum_diff_degree0 poly_shift_id poly_shift_eq cfs_times_X zero_less_Suc)
qed
lemma(in UP_cring) shift_cfs:
assumes "p ∈ carrier P"
shows "(shift k p) n = p (k + n)"
apply(induction k arbitrary: n)
by (auto simp: assms poly_shift_cfs shift_closed)
subsection‹The Derivative Operator›
definition pderiv where
"pderiv p = poly_shift (n_mult p)"
lemma pderiv_closed:
assumes "p ∈ carrier P"
shows "pderiv p ∈ carrier P"
unfolding pderiv_def
using assms n_mult_closed[of p] poly_shift_closed[of "n_mult p"]
by blast
text‹Function which obtains the first n+1 terms of f, in ascending order of degree:›
definition trms_of_deg_leq where
"trms_of_deg_leq n f ≡ f ⊖⇘(UP R)⇙ ((shift (Suc n) f) ⊗⇘UP R⇙ monom P 𝟭 (Suc n))"
lemma trms_of_deg_leq_closed:
assumes "f ∈ carrier P"
shows "trms_of_deg_leq n f ∈ carrier P"
unfolding trms_of_deg_leq_def using assms
by (metis P.m_closed P.minus_closed P_def R.one_closed monom_closed shift_closed)
lemma trms_of_deg_leq_id:
assumes "f ∈ carrier P"
shows "f ⊖⇘P⇙ (trms_of_deg_leq k f) = shift (Suc k) f ⊗⇘P⇙ monom P 𝟭 (Suc k)"
unfolding trms_of_deg_leq_def
using assms
by (smt (verit) P.add.inv_closed P.l_zero P.m_closed P.minus_add P.minus_minus P.r_neg
P_def R.one_closed UP_a_assoc a_minus_def monom_closed shift_closed)
lemma trms_of_deg_leq_id':
assumes "f ∈ carrier P"
shows "f = (trms_of_deg_leq k f) ⊕⇘P⇙ shift (Suc k) f ⊗⇘P⇙ monom P 𝟭 (Suc k)"
using trms_of_deg_leq_id assms trms_of_deg_leq_closed[of f]
by (smt (verit, ccfv_threshold) P.add.inv_closed P.l_zero P.m_closed P.minus_add P.minus_minus P.r_neg R.one_closed UP_a_assoc a_minus_def monom_closed shift_closed)
lemma deg_leqI:
assumes "p ∈ carrier P"
assumes "⋀n. n > k ⟹ p n = 𝟬"
shows "degree p ≤ k"
by (metis assms(1) assms(2) deg_zero deg_ltrm le0 le_less_linear monom_zero)
lemma deg_leE:
assumes "p ∈ carrier P"
assumes "degree p < k"
shows "p k = 𝟬"
using assms coeff_of_sum_diff_degree0 P_def coeff_simp deg_aboveD
by auto
lemma trms_of_deg_leq_deg:
assumes "f ∈ carrier P"
shows "degree (trms_of_deg_leq k f) ≤ k"
proof-
have "⋀n. (trms_of_deg_leq k f) (Suc k + n) = 𝟬"
proof-
fix n
have 0: "(shift (Suc k) f ⊗⇘UP R⇙ monom P 𝟭 (Suc k)) (Suc k + n) = shift (Suc k) f n"
using assms shift_closed cfs_monom_mult_l
by (metis P.m_comm P_def R.one_closed add.commute monom_closed times_X_pow_coeff)
then show "trms_of_deg_leq k f (Suc k + n) = 𝟬"
unfolding trms_of_deg_leq_def
using shift_cfs[of f "Suc k" n]
cfs_minus[of f "shift (Suc k) f ⊗⇘UP R⇙ monom P 𝟭 (Suc k)" "Suc k + n"]
by (metis P.m_closed P.r_neg P_def R.one_closed a_minus_def assms
cfs_minus cfs_zero monom_closed shift_closed)
qed
then show ?thesis using deg_leqI
by (metis (no_types, lifting) assms le_iff_add less_Suc_eq_0_disj less_Suc_eq_le trms_of_deg_leq_closed)
qed
lemma trms_of_deg_leq_zero_is_ctrm:
assumes "f ∈ carrier P"
assumes "degree f > 0"
shows "trms_of_deg_leq 0 f = ctrm f"
proof-
have "f = ctrm f ⊕⇘P⇙ (X ⊗⇘P⇙ (shift (Suc 0) f))"
using assms poly_shift_eq
by simp
then have "f = ctrm f ⊕⇘P⇙ (X [^]⇘UP R⇙ Suc 0 ⊗⇘P⇙ (shift (Suc 0) f))"
using P.nat_pow_eone P_def X_closed by auto
then show ?thesis
unfolding trms_of_deg_leq_def
by (metis (no_types, lifting) ctrm_is_poly One_nat_def P.add.right_cancel P.m_closed
P.minus_closed P.nat_pow_eone P_def UP_m_comm X_closed X_poly_def assms(1) shift_closed
trms_of_deg_leq_def trms_of_deg_leq_id')
qed
lemma cfs_monom_mult:
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
assumes "k < n"
shows "(p ⊗⇘P⇙ (monom P a n)) k = 𝟬"
apply(rule poly_induct3[of p])
apply (simp add: assms(1))
apply (metis (no_types, lifting) P.l_distr P.m_closed R.r_zero R.zero_closed assms(2) cfs_add monom_closed)
using assms monom_mult[of _ a _ n]
by (metis R.m_closed R.m_comm add.commute cfs_monom not_add_less1)
lemma(in UP_cring) cfs_monom_mult_2:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
assumes "m < n"
shows "((monom P a n) ⊗⇘P⇙ f) m = 𝟬"
using cfs_monom_mult
by (simp add: P.m_comm assms(1) assms(2) assms(3))
lemma trms_of_deg_leq_cfs:
assumes "f ∈ carrier P"
shows "trms_of_deg_leq n f k = (if k ≤ n then (f k) else 𝟬)"
unfolding trms_of_deg_leq_def
apply(cases "k ≤ n")
using cfs_minus[of f "shift (Suc n) f ⊗⇘UP R⇙ monom P 𝟭 (Suc n)"]
cfs_monom_mult[of _ 𝟭 k "Suc n"]
apply (metis (no_types, lifting) P.m_closed P.minus_closed P_def R.one_closed R.r_zero assms
cfs_add cfs_closed le_refl monom_closed nat_less_le nat_neq_iff not_less_eq_eq shift_closed
trms_of_deg_leq_def trms_of_deg_leq_id')
using trms_of_deg_leq_deg[of f n] deg_leE
unfolding trms_of_deg_leq_def
using assms trms_of_deg_leq_closed trms_of_deg_leq_def by auto
lemma trms_of_deg_leq_iter:
assumes "f ∈ carrier P"
shows "trms_of_deg_leq (Suc k) f = (trms_of_deg_leq k f) ⊕⇘P⇙ monom P (f (Suc k)) (Suc k)"
proof fix x
show "trms_of_deg_leq (Suc k) f x = (trms_of_deg_leq k f ⊕⇘P⇙ monom P (f (Suc k)) (Suc k)) x"
apply(cases "x ≤ k")
using trms_of_deg_leq_cfs trms_of_deg_leq_closed cfs_closed[of f "Suc k"]
cfs_add[of "trms_of_deg_leq k f" "monom P (f (Suc k)) (Suc k)" x]
apply (simp add: assms)
using deg_leE assms cfs_closed cfs_monom apply auto[1]
by (simp add: assms cfs_closed cfs_monom trms_of_deg_leq_cfs trms_of_deg_leq_closed)
qed
lemma trms_of_deg_leq_0:
assumes "f ∈ carrier P"
shows "trms_of_deg_leq 0 f = ctrm f"
by (metis One_nat_def P.r_null P_def UP_m_comm UP_zero_closed X_closed X_poly_def assms not_gr_zero
poly_shift_degree_zero shift_one trms_of_deg_leq_def trms_of_deg_leq_zero_is_ctrm trunc_simps(2) trunc_zero)
lemma trms_of_deg_leq_degree_f:
assumes "f ∈ carrier P"
shows "trms_of_deg_leq (degree f) f = f"
proof fix x
show "trms_of_deg_leq (deg R f) f x = f x"
using assms trms_of_deg_leq_cfs deg_leE[of f x]
by simp
qed
definition(in UP_cring) lin_part where
"lin_part f = trms_of_deg_leq 1 f"
lemma(in UP_cring) lin_part_id:
assumes "f ∈ carrier P"
shows "lin_part f = (ctrm f) ⊕⇘P⇙ monom P (f 1) 1"
unfolding lin_part_def
by (simp add: assms trms_of_deg_leq_0 trms_of_deg_leq_iter)
lemma(in UP_cring) lin_part_eq:
assumes "f ∈ carrier P"
shows "f = lin_part f ⊕⇘P⇙ (shift 2 f) ⊗⇘P⇙ monom P 𝟭 2"
unfolding lin_part_def
by (metis Suc_1 assms trms_of_deg_leq_id')
text‹Constant term of a substitution:›
lemma zcf_eval:
assumes "f ∈ carrier P"
shows "zcf f = to_fun f 𝟬"
using assms zcf_to_fun by blast
lemma ctrm_of_sub:
assumes "f ∈ carrier P"
assumes "g ∈ carrier P"
shows "zcf(f of g) = to_fun f (zcf g)"
apply(rule poly_induct3[of f])
apply (simp add: assms(1))
using P_def UP_cring.to_fun_closed UP_cring_axioms zcf_add zcf_to_fun assms(2) to_fun_plus sub_add sub_closed apply fastforce
using R.zero_closed zcf_to_fun assms(2) to_fun_sub monom_closed sub_closed by presburger
text‹Evaluation of linear part:›
lemma to_fun_lin_part:
assumes "f ∈ carrier P"
assumes "b ∈ carrier R"
shows "to_fun (lin_part f) b = (f 0) ⊕ (f 1) ⊗ b"
using assms lin_part_id[of f] to_fun_ctrm to_fun_monom monom_closed
by (simp add: cfs_closed to_fun_plus)
text‹Constant term of taylor expansion:›
lemma taylor_zcf:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "zcf(T⇘a⇙ f) = to_fun f a"
unfolding taylor_expansion_def
using ctrm_of_sub assms P_def zcf_eval X_plus_closed taylor_closed taylor_eval by auto
lemma(in UP_cring) taylor_eq_1:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "(T⇘a⇙ f) ⊖⇘P⇙ (trms_of_deg_leq 1 (T⇘a⇙ f)) = (shift (2::nat) (T⇘a⇙ f))⊗⇘P⇙ (X[^]⇘P⇙(2::nat))"
by (metis P.nat_pow_eone P.nat_pow_mult P_def Suc_1 taylor_closed X_closed X_poly_def assms(1)
assms(2) monom_one_Suc2 one_add_one trms_of_deg_leq_id)
lemma(in UP_cring) taylor_deg_1:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "f of (X_plus a) = (lin_part (T⇘a⇙ f)) ⊕⇘P⇙ (shift (2::nat) (T⇘a⇙ f))⊗⇘P⇙ (X[^]⇘P⇙(2::nat))"
using taylor_eq_1[of f a]
unfolding taylor_expansion_def lin_part_def
using One_nat_def X_plus_closed assms(1)
assms(2) trms_of_deg_leq_id' numeral_2_eq_2 sub_closed
by (metis P.nat_pow_Suc2 P.nat_pow_eone P_def taylor_def X_closed X_poly_def monom_one_Suc taylor_expansion_def)
lemma(in UP_cring) taylor_deg_1_eval:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
assumes "c = to_fun (shift (2::nat) (T⇘a⇙ f)) b"
assumes "fa = to_fun f a"
assumes "f'a = deriv f a"
shows "to_fun f (b ⊕ a) = fa ⊕ (f'a ⊗ b) ⊕ (c ⊗ b[^](2::nat))"
using assms taylor_deg_1 unfolding derivative_def
proof-
have 0: "to_fun f (b ⊕ a) = to_fun (f of (X_plus a)) b"
using to_fun_sub assms X_plus_closed by auto
have 1: "to_fun (lin_part (T⇘a⇙ f)) b = fa ⊕ (f'a ⊗ b) "
using assms to_fun_lin_part[of "(T⇘a⇙ f)" b]
by (metis P_def taylor_def UP_cring.taylor_zcf UP_cring.taylor_closed UP_cring_axioms zcf_def derivative_def)
have 2: "(T⇘a⇙ f) = (lin_part (T⇘a⇙ f)) ⊕⇘P⇙ ((shift 2 (T⇘a⇙ f))⊗⇘P⇙X[^]⇘P⇙(2::nat))"
using lin_part_eq[of "(T⇘a⇙f)"] assms(1) assms(2) taylor_closed
by (metis taylor_def taylor_deg_1 taylor_expansion_def)
then have "to_fun (T⇘a⇙f) b = fa ⊕ (f'a ⊗ b) ⊕ to_fun ((shift 2 (T⇘a⇙ f))⊗⇘P⇙X[^]⇘P⇙(2::nat)) b"
using 1 2
by (metis P.nat_pow_closed taylor_closed UP_mult_closed X_closed assms(1) assms(2) assms(3)
to_fun_plus lin_part_def shift_closed trms_of_deg_leq_closed)
then have "to_fun (T⇘a⇙f) b = fa ⊕ (f'a ⊗ b) ⊕ c ⊗ to_fun (X[^]⇘P⇙(2::nat)) b"
by (simp add: taylor_closed X_closed assms(1) assms(2) assms(3) assms(4) to_fun_mult shift_closed)
then have 3: "to_fun f (b ⊕ a)= fa ⊕ (f'a ⊗ b) ⊕ c ⊗ to_fun (X[^]⇘P⇙(2::nat)) b"
using taylor_eval assms(1) assms(2) assms(3) by auto
have "to_fun (X[^]⇘P⇙(2::nat)) b = b[^](2::nat)"
by (metis P.nat_pow_Suc2 P.nat_pow_eone R.nat_pow_Suc2
R.nat_pow_eone Suc_1 to_fun_X
X_closed assms(3) to_fun_mult)
then show ?thesis
using 3 by auto
qed
lemma(in UP_cring) taylor_deg_1_eval':
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
assumes "c = to_fun (shift (2::nat) (T⇘a⇙ f)) b"
assumes "fa = to_fun f a"
assumes "f'a = deriv f a"
shows "to_fun f (a ⊕ b) = fa ⊕ (f'a ⊗ b) ⊕ (c ⊗ b[^](2::nat))"
using R.add.m_comm taylor_deg_1_eval assms(1) assms(2) assms(3) assms(4) assms(5) assms(6)
by auto
lemma(in UP_cring) taylor_deg_1_eval'':
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
assumes "c = to_fun (shift (2::nat) (T⇘a⇙ f)) (⊖b)"
shows "to_fun f (a ⊖ b) = (to_fun f a) ⊖ (deriv f a ⊗ b) ⊕ (c ⊗ b[^](2::nat))"
proof-
have "⊖b ∈ carrier R"
using assms
by blast
then have 0: "to_fun f (a ⊖ b) = (to_fun f a)⊕ (deriv f a ⊗ (⊖b)) ⊕ (c ⊗ (⊖b)[^](2::nat))"
unfolding a_minus_def
using taylor_deg_1_eval'[of f a "⊖b" c "(to_fun f a)" "deriv f a"] assms
by auto
have 1: "⊖ (deriv f a ⊗ b) = (deriv f a ⊗ (⊖b))"
using assms
by (simp add: R.r_minus deriv_closed)
have 2: "(c ⊗ b[^](2::nat)) = (c ⊗ (⊖b)[^](2::nat))"
using assms
by (metis R.add.inv_closed R.add.inv_solve_right R.l_zero R.nat_pow_Suc2
R.nat_pow_eone R.zero_closed Suc_1 UP_ring_axioms UP_ring_def
ring.ring_simprules(26) ring.ring_simprules(27))
show ?thesis
using 0 1 2
unfolding a_minus_def
by simp
qed
lemma(in UP_cring) taylor_deg_1_expansion:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
assumes "c = to_fun (shift (2::nat) (T⇘a⇙ f)) (b ⊖ a)"
assumes "fa = to_fun f a"
assumes "f'a = deriv f a"
shows "to_fun f (b) = fa ⊕ f'a ⊗ (b ⊖ a) ⊕ (c ⊗ (b ⊖ a)[^](2::nat))"
proof-
obtain b' where b'_def: "b'= b ⊖ a "
by simp
then have b'_def': "b = b' ⊕ a"
using assms
by (metis R.add.inv_solve_right R.minus_closed R.minus_eq)
have "to_fun f (b' ⊕ a) = fa ⊕ (f'a ⊗ b') ⊕ (c ⊗ b'[^](2::nat))"
using assms taylor_deg_1_eval[of f a b' c fa f'a] b'_def
by blast
then have "to_fun f (b) = fa ⊕ (f'a ⊗ b') ⊕ (c ⊗ b'[^](2::nat))"
using b'_def'
by auto
then show "to_fun f (b) = fa ⊕ f'a ⊗ (b ⊖ a) ⊕ c ⊗ (b ⊖ a) [^] (2::nat)"
using b'_def
by auto
qed
lemma(in UP_cring) Taylor_deg_1_expansion':
assumes "f ∈ carrier (UP R)"
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "∃c ∈ carrier R. to_fun f (b) = (to_fun f a) ⊕ (deriv f a) ⊗ (b ⊖ a) ⊕ (c ⊗ (b ⊖ a)[^](2::nat))"
using taylor_deg_1_expansion[of f a b] assms unfolding P_def
by (metis P_def R.minus_closed taylor_closed shift_closed to_fun_closed)
text‹Basic Properties of deriv and pderiv:›
lemma n_mult_degree_bound:
assumes "f ∈ carrier P"
shows "degree (n_mult f) ≤ degree f"
apply(rule deg_leqI)
apply (simp add: assms n_mult_closed)
by (simp add: assms deg_leE n_mult_def)
lemma pderiv_deg_0[simp]:
assumes "f ∈ carrier P"
assumes "degree f = 0"
shows "pderiv f = 𝟬⇘P⇙"
proof-
have "degree (n_mult f) = 0"
using P_def n_mult_degree_bound assms(1) assms(2) by fastforce
then show ?thesis
unfolding pderiv_def
by (simp add: assms(1) n_mult_closed poly_shift_degree_zero)
qed
lemma deriv_deg_0:
assumes "f ∈ carrier P"
assumes "degree f = 0"
assumes "a ∈ carrier R"
shows "deriv f a = 𝟬"
unfolding derivative_def taylor_expansion_def
using X_plus_closed assms(1) assms(2) assms(3) deg_leE sub_const by force
lemma poly_shift_monom':
assumes "a ∈ carrier R"
shows "poly_shift (a ⊙⇘P⇙ (X[^]⇘P⇙(Suc n))) = a⊙⇘P⇙(X[^]⇘P⇙n)"
using assms monom_rep_X_pow poly_shift_monom by auto
lemma monom_coeff:
assumes "a ∈ carrier R"
shows "(a ⊙⇘P⇙ X [^]⇘P⇙ (n::nat)) k = (if (k = n) then a else 𝟬)"
using assms cfs_monom monom_rep_X_pow by auto
lemma cfs_n_mult:
assumes "p ∈ carrier P"
shows "n_mult p n = [n]⋅(p n)"
by (simp add: n_mult_def)
lemma cfs_add_nat_pow:
assumes "p ∈ carrier P"
shows "([(n::nat)]⋅⇘P⇙p) k = [n]⋅(p k)"
apply(induction n) by (auto simp: assms)
lemma cfs_add_int_pow:
assumes "p ∈ carrier P"
shows "([(n::int)]⋅⇘P⇙p) k = [n]⋅(p k)"
apply(induction n)
by(auto simp: add_pow_int_ge assms cfs_add_nat_pow add_pow_int_lt)
lemma add_nat_pow_monom:
assumes "a ∈ carrier R"
shows "[(n::nat)]⋅⇘P⇙monom P a k = monom P ([n]⋅a) k"
apply(rule ext)
by (simp add: assms cfs_add_nat_pow cfs_monom)
lemma add_int_pow_monom:
assumes "a ∈ carrier R"
shows "[(n::int)]⋅⇘P⇙monom P a k = monom P ([n]⋅a) k"
apply(rule ext)
by (simp add: assms cfs_add_int_pow cfs_monom)
lemma n_mult_monom:
assumes "a ∈ carrier R"
shows "n_mult (monom P a (Suc n)) = monom P ([Suc n]⋅a) (Suc n)"
apply(rule ext)
unfolding n_mult_def
using assms cfs_monom by auto
lemma pderiv_monom:
assumes "a ∈ carrier R"
shows "pderiv (monom P a n) = monom P ([n]⋅a) (n-1)"
apply(cases "n = 0")
apply (simp add: assms)
unfolding pderiv_def
using assms Suc_diff_1[of n] n_mult_monom[of a "n-1"] poly_shift_monom[of "[Suc (n-1)]⋅a" "Suc (n-1)"]
by (metis R.add.nat_pow_closed neq0_conv poly_shift_monom)
lemma pderiv_monom':
assumes "a ∈ carrier R"
shows "pderiv (a ⊙⇘P⇙ X[^]⇘P⇙(n::nat)) = ([n]⋅a)⊙⇘P⇙ X[^]⇘P⇙(n-1)"
using assms pderiv_monom[of a n ]
by (simp add: P_def UP_cring.monom_rep_X_pow UP_cring_axioms)
lemma n_mult_add:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
shows "n_mult (p ⊕⇘P⇙ q) = n_mult p ⊕⇘P⇙ n_mult q"
proof(rule ext) fix x show "n_mult (p ⊕⇘P⇙ q) x = (n_mult p ⊕⇘P⇙ n_mult q) x"
using assms R.add.nat_pow_distrib[of "p x" "q x" x] cfs_add[of p q x]
cfs_add[of "n_mult p" "n_mult q" x] n_mult_closed
unfolding n_mult_def
by (simp add: cfs_closed)
qed
lemma pderiv_add:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
shows "pderiv (p ⊕⇘P⇙ q) = pderiv p ⊕⇘P⇙ pderiv q"
unfolding pderiv_def
using assms poly_shift_add n_mult_add
by (simp add: n_mult_closed)
lemma zcf_monom_sub:
assumes "p ∈ carrier P"
shows "zcf ((monom P 𝟭 (Suc n)) of p) = zcf p [^] (Suc n)"
apply(induction n)
using One_nat_def P.nat_pow_eone R.nat_pow_eone R.one_closed R.zero_closed zcf_to_fun assms to_fun_closed monom_sub smult_one apply presburger
using P_def UP_cring.ctrm_of_sub UP_cring_axioms zcf_to_fun assms to_fun_closed to_fun_monom monom_closed
by fastforce
lemma zcf_monom_sub':
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
shows "zcf ((monom P a (Suc n)) of p) = a ⊗ zcf p [^] (Suc n)"
using zcf_monom_sub assms P_def R.zero_closed UP_cring.ctrm_of_sub UP_cring.to_fun_monom UP_cring_axioms
zcf_to_fun to_fun_closed monom_closed by fastforce
lemma deriv_monom:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "deriv (monom P a n) b = ([n]⋅a)⊗(b[^](n-1))"
proof(induction n)
case 0
have 0: "b [^] ((0::nat) - 1) ∈ carrier R"
using assms
by simp
then show ?case unfolding derivative_def using assms
by (metis R.add.nat_pow_0 R.l_null deg_const deriv_deg_0 derivative_def monom_closed)
next
case (Suc n)
show ?case
proof(cases "n = 0")
case True
have T0: "[Suc n] ⋅ a ⊗ b [^] (Suc n - 1) = a"
by (simp add: True assms(1))
have T1: "(X_poly R ⊕⇘UP R⇙ to_polynomial R b) [^]⇘UP R⇙ Suc n = X_poly R ⊕⇘UP R⇙ to_polynomial R b "
using P.nat_pow_eone P_def True UP_a_closed X_closed assms(2) to_poly_closed by auto
then show ?thesis
unfolding derivative_def taylor_expansion_def
using T0 T1 True sub_monom(2)[of "X_plus b" a "Suc n"] cfs_add assms
unfolding P_def X_poly_plus_def to_polynomial_def X_poly_def
by (metis One_nat_def P_def R.add.nat_pow_eone R.nat_pow_0 UP_cring.cfs_X_plus X_plus_closed X_poly_def X_poly_plus_def cfs_smult diff_Suc_1' is_UP_cring n_not_Suc_n to_polynomial_def)
next
case False
have "deriv (monom P a (Suc n)) b = ((monom P a (Suc n)) of (X_plus b)) 1"
unfolding derivative_def taylor_expansion_def
by auto
then have "deriv (monom P a (Suc n)) b = (((monom P a n) of (X_plus b)) ⊗⇘P⇙ (X_plus b)) 1"
using monom_mult[of a 𝟭 n 1] sub_mult[of "X_plus b" "monom P a n" "monom P 𝟭 1" ] X_plus_closed[of b] assms
by (metis lcf_monom(1) P.l_one P.nat_pow_eone P_def R.one_closed R.r_one Suc_eq_plus1
deg_one monom_closed monom_one sub_monom(1) to_poly_inverse)
then have "deriv (monom P a (Suc n)) b = (((monom P a n) of (X_plus b)) ⊗⇘P⇙ (monom P 𝟭 1) ⊕⇘P⇙
(((monom P a n) of (X_plus b)) ⊗⇘P⇙ to_poly b)) 1"
unfolding X_poly_plus_def
by (metis P.r_distr P_def X_closed X_plus_closed X_poly_def X_poly_plus_def assms(1) assms(2) monom_closed sub_closed to_poly_closed)
then have "deriv (monom P a (Suc n)) b = ((monom P a n) of (X_plus b)) 0 ⊕ b ⊗ ((monom P a n) of (X_plus b)) 1"
unfolding X_poly_plus_def
by (smt (verit) One_nat_def P.m_closed P_def UP_m_comm X_closed X_plus_closed X_poly_def X_poly_plus_def
assms(1) assms(2) cfs_add cfs_monom_mult_l monom_closed plus_1_eq_Suc sub_closed cfs_times_X to_polynomial_def)
then have "deriv (monom P a (Suc n)) b = ((monom P a n) of (X_plus b)) 0 ⊕ b ⊗ (deriv (monom P a n) b)"
by (simp add: derivative_def taylor_expansion_def)
then have "deriv (monom P a (Suc n)) b = ((monom P a n) of (X_plus b)) 0 ⊕ b ⊗ ( ([n]⋅a)⊗(b[^](n-1)))"
by (simp add: Suc)
then have 0: "deriv (monom P a (Suc n)) b = ((monom P a n) of (X_plus b)) 0 ⊕ ([n]⋅a)⊗(b[^]n)"
using assms R.m_comm[of b] R.nat_pow_mult[of b "n-1" 1] False
by (metis (no_types, lifting) R.add.nat_pow_closed R.m_lcomm R.nat_pow_closed R.nat_pow_eone add.commute add_eq_if plus_1_eq_Suc)
have 1: "((monom P a n) of (X_plus b)) 0 = a ⊗ b[^]n"
unfolding X_poly_plus_def using zcf_monom_sub'
by (smt (verit) ctrm_of_sub One_nat_def P_def R.l_zero R.one_closed UP_cring.zcf_to_poly
UP_cring.f_minus_ctrm UP_cring_axioms X_plus_closed X_poly_def X_poly_plus_def zcf_add
zcf_def assms(1) assms(2) to_fun_monom monom_closed monom_one_Suc2 poly_shift_id poly_shift_monom to_poly_closed)
show ?thesis
using 0 1 R.add.nat_pow_Suc2 R.add.nat_pow_closed R.l_distr R.nat_pow_closed assms(1) assms(2) diff_Suc_1 by presburger
qed
qed
lemma deriv_smult:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
assumes "g ∈ carrier P"
shows "deriv (a ⊙⇘P⇙ g) b = a ⊗ (deriv g b)"
unfolding derivative_def taylor_expansion_def
using assms sub_smult X_plus_closed cfs_smult
by (simp add: sub_closed)
lemma deriv_const:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "deriv (monom P a 0) b = 𝟬"
unfolding derivative_def
using assms taylor_closed taylor_def taylor_deg deg_leE by auto
lemma deriv_monom_deg_one:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "deriv (monom P a 1) b = a"
unfolding derivative_def taylor_expansion_def
using assms cfs_X_plus[of b 1] sub_monom_deg_one X_plus_closed[of b]
by simp
lemma monom_Suc:
assumes "a ∈ carrier R"
shows "monom P a (Suc n) = monom P 𝟭 1 ⊗⇘P⇙ monom P a n"
"monom P a (Suc n) = monom P a n ⊗⇘P⇙ monom P 𝟭 1"
apply (metis R.l_one R.one_closed Suc_eq_plus1_left assms monom_mult)
by (metis R.one_closed R.r_one Suc_eq_plus1 assms monom_mult)
subsection‹The Product Rule›
lemma(in UP_cring) times_x_product_rule:
assumes "f ∈ carrier P"
shows "pderiv (f ⊗⇘P⇙ up_ring.monom P 𝟭 1) = f ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P 𝟭 1"
proof(rule poly_induct3[of f])
show "f ∈ carrier P"
using assms by blast
show "⋀p q. q ∈ carrier P ⟹
p ∈ carrier P ⟹
pderiv (p ⊗⇘P⇙ up_ring.monom P 𝟭 1) = p ⊕⇘P⇙ pderiv p ⊗⇘P⇙ up_ring.monom P 𝟭 1 ⟹
pderiv (q ⊗⇘P⇙ up_ring.monom P 𝟭 1) = q ⊕⇘P⇙ pderiv q ⊗⇘P⇙ up_ring.monom P 𝟭 1 ⟹
pderiv ((p ⊕⇘P⇙ q) ⊗⇘P⇙ up_ring.monom P 𝟭 1) = p ⊕⇘P⇙ q ⊕⇘P⇙ pderiv (p ⊕⇘P⇙ q) ⊗⇘P⇙ up_ring.monom P 𝟭 1"
proof- fix p q assume A: "q ∈ carrier P"
"p ∈ carrier P"
"pderiv (p ⊗⇘P⇙ up_ring.monom P 𝟭 1) = p ⊕⇘P⇙ pderiv p ⊗⇘P⇙ up_ring.monom P 𝟭 1"
"pderiv (q ⊗⇘P⇙ up_ring.monom P 𝟭 1) = q ⊕⇘P⇙ pderiv q ⊗⇘P⇙ up_ring.monom P 𝟭 1"
have 0: "(p ⊕⇘P⇙ q) ⊗⇘P⇙ up_ring.monom P 𝟭 1 = (p ⊗⇘P⇙ up_ring.monom P 𝟭 1) ⊕⇘P⇙ (q ⊗⇘P⇙ up_ring.monom P 𝟭 1)"
using A assms by (meson R.one_closed UP_l_distr is_UP_monomE(1) is_UP_monomI)
have 1: "pderiv ((p ⊕⇘P⇙ q) ⊗⇘P⇙ up_ring.monom P 𝟭 1) = pderiv (p ⊗⇘P⇙ up_ring.monom P 𝟭 1) ⊕⇘P⇙ pderiv (q ⊗⇘P⇙ up_ring.monom P 𝟭 1)"
unfolding 0 apply(rule pderiv_add)
using A is_UP_monomE(1) monom_is_UP_monom(1) apply blast
using A is_UP_monomE(1) monom_is_UP_monom(1) by blast
have 2: "pderiv ((p ⊕⇘P⇙ q) ⊗⇘P⇙ up_ring.monom P 𝟭 1) = p ⊕⇘P⇙ pderiv p ⊗⇘P⇙ up_ring.monom P 𝟭 1 ⊕⇘P⇙ (q ⊕⇘P⇙ pderiv q ⊗⇘P⇙ up_ring.monom P 𝟭 1)"
unfolding 1 A by blast
have 3: "pderiv ((p ⊕⇘P⇙ q) ⊗⇘P⇙ up_ring.monom P 𝟭 1) = p ⊕⇘P⇙ q ⊕⇘P⇙ (pderiv p ⊗⇘P⇙ up_ring.monom P 𝟭 1 ⊕⇘P⇙ pderiv q ⊗⇘P⇙ up_ring.monom P 𝟭 1)"
unfolding 2
using A P.add.m_lcomm R.one_closed UP_a_assoc UP_a_closed UP_mult_closed is_UP_monomE(1) monom_is_UP_monom(1) pderiv_closed by presburger
have 4: "pderiv ((p ⊕⇘P⇙ q) ⊗⇘P⇙ up_ring.monom P 𝟭 1) = p ⊕⇘P⇙ q ⊕⇘P⇙ ((pderiv p ⊕⇘P⇙ pderiv q) ⊗⇘P⇙ up_ring.monom P 𝟭 1)"
unfolding 3 using A P.l_distr R.one_closed is_UP_monomE(1) monom_is_UP_monom(1) pderiv_closed by presburger
show 5: "pderiv ((p ⊕⇘P⇙ q) ⊗⇘P⇙ up_ring.monom P 𝟭 1) = p ⊕⇘P⇙ q ⊕⇘P⇙ pderiv (p ⊕⇘P⇙ q) ⊗⇘P⇙ up_ring.monom P 𝟭 1"
unfolding 4 using pderiv_add A by presburger
qed
show "⋀a n. a ∈ carrier R ⟹
pderiv (up_ring.monom P a n ⊗⇘P⇙ up_ring.monom P 𝟭 1) = up_ring.monom P a n ⊕⇘P⇙ pderiv (up_ring.monom P a n) ⊗⇘P⇙ up_ring.monom P 𝟭 1"
proof- fix a n assume A: "a ∈ carrier R"
have 0: "up_ring.monom P a n ⊗⇘P⇙ up_ring.monom P 𝟭 1 = up_ring.monom P a (Suc n)"
using A monom_Suc(2) by presburger
have 1: "pderiv (up_ring.monom P a n ⊗⇘P⇙ up_ring.monom P 𝟭 1) = [(Suc n)] ⋅⇘P⇙ (up_ring.monom P a n)"
unfolding 0 using A add_nat_pow_monom n_mult_monom pderiv_def poly_shift_monom
by (simp add: P_def)
have 2: "pderiv (up_ring.monom P a n ⊗⇘P⇙ up_ring.monom P 𝟭 1) = (up_ring.monom P a n) ⊕⇘P⇙ [n] ⋅⇘P⇙ (up_ring.monom P a n)"
unfolding 1 using A P.add.nat_pow_Suc2 is_UP_monomE(1) monom_is_UP_monom(1) by blast
have 3: "pderiv (up_ring.monom P a n) ⊗⇘P⇙ up_ring.monom P 𝟭 1 = [n] ⋅⇘P⇙ (up_ring.monom P a n)"
apply(cases "n = 0")
using A add_nat_pow_monom n_mult_monom pderiv_def poly_shift_monom pderiv_deg_0 apply auto[1]
using monom_Suc(2)[of a "n-1"] A add_nat_pow_monom n_mult_monom pderiv_def poly_shift_monom
by (metis R.add.nat_pow_closed Suc_eq_plus1 add_eq_if monom_Suc(2) pderiv_monom)
show "pderiv (up_ring.monom P a n ⊗⇘P⇙ up_ring.monom P 𝟭 1) = up_ring.monom P a n ⊕⇘P⇙ pderiv (up_ring.monom P a n) ⊗⇘P⇙ up_ring.monom P 𝟭 1"
unfolding 2 3 by blast
qed
qed
lemma(in UP_cring) deg_one_eval:
assumes "g ∈ carrier (UP R)"
assumes "deg R g = 1"
shows "⋀t. t ∈ carrier R ⟹ to_fun g t = g 0 ⊕ (g 1)⊗t"
proof-
obtain h where h_def: "h = ltrm g"
by blast
have 0: "deg R (g ⊖⇘UP R⇙ h) = 0"
using assms unfolding h_def
by (metis ltrm_closed ltrm_eq_imp_deg_drop ltrm_monom P_def UP_car_memE(1) less_one)
have 1: "g ⊖⇘UP R⇙ h = to_poly (g 0)"
proof(rule ext) fix x show "(g ⊖⇘UP R⇙ h) x = to_polynomial R (g 0) x"
proof(cases "x = 0")
case True
have T0: "h 0 = 𝟬"
unfolding h_def using assms UP_car_memE(1) cfs_monom by presburger
have T1: "(g ⊖⇘UP R⇙ h) 0 = g 0 ⊖ h 0"
using ltrm_closed P_def assms(1) cfs_minus h_def by blast
then show ?thesis using T0 assms
by (smt (verit) "0" ltrm_closed ltrm_deg_0 P.minus_closed P_def UP_car_memE(1) UP_zero_closed zcf_def zcf_zero deg_zero degree_to_poly h_def to_poly_closed to_poly_inverse to_poly_minus trunc_simps(2) trunc_zero)
next
case False
then have "x > 0"
by presburger
then show ?thesis
by (metis "0" ltrm_closed P.minus_closed P_def UP_car_memE(1) UP_cring.degree_to_poly UP_cring_axioms assms(1) deg_leE h_def to_poly_closed)
qed
qed
have 2: "g = (g ⊖⇘UP R⇙ h) ⊕⇘UP R⇙ h"
unfolding h_def using assms
by (metis "1" P_def h_def lin_part_def lin_part_id to_polynomial_def trms_of_deg_leq_degree_f)
fix t assume A: "t ∈ carrier R"
have 3: " to_fun g t = to_fun (g ⊖⇘UP R⇙ h) t ⊕ to_fun h t"
using 2
by (metis "1" A P_def UP_car_memE(1) assms(1) h_def monom_closed to_fun_plus to_polynomial_def)
then show "to_fun g t = g 0 ⊕ g 1 ⊗ t "
unfolding 1 h_def
using A P_def UP_cring.lin_part_def UP_cring_axioms assms(1) assms(2) to_fun_lin_part trms_of_deg_leq_degree_f by fastforce
qed
lemma nmult_smult:
assumes "a ∈ carrier R"
assumes "f ∈ carrier P"
shows "n_mult (a ⊙⇘P⇙ f) = a ⊙⇘P⇙ (n_mult f)"
apply(rule poly_induct4[of f])
apply (simp add: assms(2))
using assms(1) n_mult_add n_mult_closed smult_closed smult_r_distr apply presburger
using assms apply(intro ext, metis (no_types, lifting) ctrm_smult ltrm_deg_0 P_def R.add.nat_pow_0 UP_cring.ctrm_degree UP_cring.n_mult_closed UP_cring.n_mult_def UP_cring_axioms UP_smult_closed UP_zero_closed zcf_degree_zero zcf_zero deg_const deg_zero le_0_eq monom_closed n_mult_degree_bound smult_r_null)
using monom_mult_smult n_mult_monom assms
by (smt (verit) lcf_monom(1) P_def R.add.nat_pow_closed R.add_pow_rdistr R.zero_closed UP_cring.to_poly_mult_simp(1) UP_cring_axioms UP_smult_closed cfs_closed cring_lcf_mult monom_closed to_polynomial_def)
lemma pderiv_smult:
assumes "a ∈ carrier R"
assumes "f ∈ carrier P"
shows "pderiv (a ⊙⇘P⇙ f) = a ⊙⇘P⇙ (pderiv f)"
unfolding pderiv_def
using assms
by (simp add: n_mult_closed nmult_smult poly_shift_s_mult)
lemma(in UP_cring) pderiv_minus:
assumes "a ∈ carrier P"
assumes "b ∈ carrier P"
shows "pderiv (a ⊖⇘P⇙ b) = pderiv a ⊖⇘P⇙ pderiv b"
proof-
have "⊖⇘P⇙ b = (⊖𝟭)⊙⇘P⇙b"
using R.one_closed UP_smult_one assms(2) smult_l_minus by presburger
thus ?thesis unfolding a_minus_def using pderiv_add assms pderiv_smult
by (metis P.add.inv_closed R.add.inv_closed R.one_closed UP_smult_one pderiv_closed smult_l_minus)
qed
lemma(in UP_cring) pderiv_const:
assumes "b ∈ carrier R"
shows "pderiv (up_ring.monom P b 0) = 𝟬⇘P⇙"
using assms pderiv_monom[of b 0] deg_const is_UP_monomE(1) monom_is_UP_monom(1) pderiv_deg_0
by blast
lemma(in UP_cring) pderiv_minus_const:
assumes "a ∈ carrier P"
assumes "b ∈ carrier R"
shows "pderiv (a ⊖⇘P⇙ up_ring.monom P b 0) = pderiv a"
using pderiv_minus[of a "up_ring.monom P b 0" ] assms pderiv_const[of b]
by (smt (verit) P.l_zero P.minus_closed P_def UP_cring.pderiv_const UP_cring.pderiv_minus UP_cring.poly_shift_eq UP_cring_axioms cfs_closed monom_closed pderiv_add pderiv_closed poly_shift_id)
lemma(in UP_cring) monom_product_rule:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "pderiv (f ⊗⇘P⇙ up_ring.monom P a n) = f ⊗⇘P⇙ pderiv (up_ring.monom P a n) ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P a n"
proof-
have "∀f. f ∈ carrier P ⟶ pderiv (f ⊗⇘P⇙ up_ring.monom P a n) = f ⊗⇘P⇙ pderiv (up_ring.monom P a n) ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P a n"
proof(induction n)
case 0
show ?case
proof fix f show "f ∈ carrier P ⟶ pderiv (f ⊗⇘P⇙ up_ring.monom P a 0) = f ⊗⇘P⇙ pderiv (up_ring.monom P a 0) ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P a 0 "
proof assume A: "f ∈ carrier P"
have 0: "f ⊗⇘P⇙ up_ring.monom P a 0 = a ⊙⇘P⇙f"
using assms A UP_m_comm is_UP_monomE(1) monom_is_UP_monom(1) monom_mult_is_smult by presburger
have 1: "f ⊗⇘P⇙ pderiv (up_ring.monom P a 0) = 𝟬⇘P⇙"
using A assms P.r_null pderiv_const by presburger
have 2: "pderiv f ⊗⇘P⇙ up_ring.monom P a 0 = a ⊙⇘P⇙ pderiv f"
using assms A UP_m_comm is_UP_monomE(1) monom_is_UP_monom(1) monom_mult_is_smult pderiv_closed by presburger
show "pderiv (f ⊗⇘P⇙ up_ring.monom P a 0) = f ⊗⇘P⇙ pderiv (up_ring.monom P a 0) ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P a 0"
unfolding 0 1 2 using A UP_l_zero UP_smult_closed assms(2) pderiv_closed pderiv_smult by presburger
qed
qed
next
case (Suc n)
show "∀f. f ∈ carrier P ⟶
pderiv (f ⊗⇘P⇙ up_ring.monom P a (Suc n)) = f ⊗⇘P⇙ pderiv (up_ring.monom P a (Suc n)) ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P a (Suc n)"
proof fix f
show "f ∈ carrier P ⟶
pderiv (f ⊗⇘P⇙ up_ring.monom P a (Suc n)) = f ⊗⇘P⇙ pderiv (up_ring.monom P a (Suc n)) ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P a (Suc n)"
proof
assume A: "f ∈ carrier P"
show " pderiv (f ⊗⇘P⇙ up_ring.monom P a (Suc n)) = f ⊗⇘P⇙ pderiv (up_ring.monom P a (Suc n)) ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P a (Suc n)"
proof(cases "n = 0")
case True
have 0: "(f ⊗⇘P⇙ up_ring.monom P a (Suc n)) = a ⊙⇘P⇙ f ⊗⇘P⇙ up_ring.monom P 𝟭 1"
proof -
have "∀n. up_ring.monom P a n ∈ carrier P"
using assms(2) is_UP_monomE(1) monom_is_UP_monom(1) by presburger
then show ?thesis
by (metis A P.m_assoc P.m_comm R.one_closed True assms(2) is_UP_monomE(1) monom_Suc(2) monom_is_UP_monom(1) monom_mult_is_smult)
qed
have 1: "f ⊗⇘P⇙ pderiv (up_ring.monom P a (Suc n)) = a ⊙⇘P⇙ f"
using assms True
by (metis A One_nat_def P.m_comm R.add.nat_pow_eone diff_Suc_1 is_UP_monomE(1) is_UP_monomI monom_mult_is_smult pderiv_monom)
have 2: "pderiv f ⊗⇘P⇙ up_ring.monom P a (Suc n) = a ⊙⇘P⇙ (pderiv f ⊗⇘P⇙ up_ring.monom P 𝟭 1)"
using A assms unfolding True
by (metis P.m_lcomm R.one_closed UP_mult_closed is_UP_monomE(1) monom_Suc(2) monom_is_UP_monom(1) monom_mult_is_smult pderiv_closed)
have 3: "a ⊙⇘P⇙ f ⊕⇘P⇙ a ⊙⇘P⇙ (pderiv f ⊗⇘P⇙ up_ring.monom P 𝟭 1) = a ⊙⇘P⇙ (f ⊕⇘P⇙(pderiv f ⊗⇘P⇙ up_ring.monom P 𝟭 1))"
using assms A P.m_closed R.one_closed is_UP_monomE(1) monom_is_UP_monom(1) pderiv_closed smult_r_distr by presburger
show ?thesis
unfolding 0 1 2 3
using A times_x_product_rule P.m_closed R.one_closed UP_smult_assoc2 assms(2) is_UP_monomE(1) monom_is_UP_monom(1) pderiv_smult by presburger
next
case False
have IH: "pderiv ((f ⊗⇘P⇙up_ring.monom P 𝟭 1) ⊗⇘P⇙ up_ring.monom P a n) = (f ⊗⇘P⇙up_ring.monom P 𝟭 1) ⊗⇘P⇙ pderiv (up_ring.monom P a n) ⊕⇘P⇙ pderiv (f ⊗⇘P⇙up_ring.monom P 𝟭 1) ⊗⇘P⇙ up_ring.monom P a n"
using Suc A P.m_closed R.one_closed is_UP_monomE(1) is_UP_monomI by presburger
have 0: "f ⊗⇘P⇙ up_ring.monom P a (Suc n) = (f ⊗⇘P⇙up_ring.monom P 𝟭 1) ⊗⇘P⇙ up_ring.monom P a n"
using A R.one_closed UP_m_assoc assms(2) is_UP_monomE(1) monom_Suc(1) monom_is_UP_monom(1) by presburger
have 1: "(f ⊗⇘P⇙up_ring.monom P 𝟭 1) ⊗⇘P⇙ pderiv (up_ring.monom P a n) ⊕⇘P⇙ pderiv (f ⊗⇘P⇙up_ring.monom P 𝟭 1) ⊗⇘P⇙ up_ring.monom P a n =
(f ⊗⇘P⇙up_ring.monom P 𝟭 1) ⊗⇘P⇙ pderiv (up_ring.monom P a n) ⊕⇘P⇙ (f ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P 𝟭 1) ⊗⇘P⇙ up_ring.monom P a n "
using A times_x_product_rule by presburger
have 2: "(f ⊗⇘P⇙up_ring.monom P 𝟭 1) ⊗⇘P⇙ pderiv (up_ring.monom P a n) =(f ⊗⇘P⇙up_ring.monom P ([n]⋅a) n)"
proof-
have 20: "up_ring.monom P ([n] ⋅ a) (n) = up_ring.monom P 𝟭 1 ⊗⇘P⇙ up_ring.monom P ([n] ⋅ a) (n - 1)"
using A assms False monom_mult[of 𝟭 "[n]⋅a" 1 "n-1"]
by (metis R.add.nat_pow_closed R.l_one R.one_closed Suc_eq_plus1 add.commute add_eq_if )
show ?thesis unfolding 20 using assms A False pderiv_monom[of a n]
using P.m_assoc R.one_closed is_UP_monomE(1) monom_is_UP_monom(1) by simp
qed
have 3: "(f ⊗⇘P⇙up_ring.monom P ([n]⋅a) n) = [n]⋅⇘P⇙(f ⊗⇘P⇙up_ring.monom P a n)"
using A assms by (metis P.add_pow_rdistr add_nat_pow_monom is_UP_monomE(1) monom_is_UP_monom(1))
have 4: "pderiv (f ⊗⇘P⇙ up_ring.monom P 𝟭 1) = (f ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P 𝟭 1)"
using times_x_product_rule A by blast
have 5: " (f ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P 𝟭 1) ⊗⇘P⇙ up_ring.monom P a n =
(f ⊗⇘P⇙ up_ring.monom P a n ) ⊕⇘P⇙ (pderiv f ⊗⇘P⇙ up_ring.monom P 𝟭 1 ⊗⇘P⇙ up_ring.monom P a n )"
using A assms by (meson P.l_distr P.m_closed R.one_closed is_UP_monomE(1) is_UP_monomI pderiv_closed)
have 6: " (f ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P 𝟭 1) ⊗⇘P⇙ up_ring.monom P a n =
(f ⊗⇘P⇙ up_ring.monom P a n ) ⊕⇘P⇙ (pderiv f ⊗⇘P⇙ up_ring.monom P 𝟭 1 ⊗⇘P⇙ up_ring.monom P a n )"
using A assms False 5 by blast
have 7: "(f ⊗⇘P⇙up_ring.monom P 𝟭 1) ⊗⇘P⇙ pderiv (up_ring.monom P a n) ⊕⇘P⇙ pderiv (f ⊗⇘P⇙up_ring.monom P 𝟭 1) ⊗⇘P⇙ up_ring.monom P a n =
[(Suc n)] ⋅⇘P⇙ (f ⊗⇘P⇙ up_ring.monom P a n) ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P 𝟭 1 ⊗⇘P⇙ up_ring.monom P a n"
unfolding 2 3 5 6 using assms A P.a_assoc
by (smt (verit) "1" "2" "3" "6" P.add.nat_pow_Suc P.m_closed R.one_closed is_UP_monomE(1) monom_is_UP_monom(1) pderiv_closed)
have 8: "pderiv (f ⊗⇘P⇙ up_ring.monom P a (Suc n)) = pderiv ((f ⊗⇘P⇙up_ring.monom P 𝟭 1) ⊗⇘P⇙ up_ring.monom P a n)"
using A assms 0 by presburger
show " pderiv (f ⊗⇘P⇙ up_ring.monom P a (Suc n)) = f ⊗⇘P⇙ pderiv (up_ring.monom P a (Suc n)) ⊕⇘P⇙ pderiv f ⊗⇘P⇙ up_ring.monom P a (Suc n)"
unfolding 8 IH 0 1 2 3 4 5 6
by (smt (verit) "2" "4" "6" "7" A P.add_pow_rdistr R.one_closed UP_m_assoc add_nat_pow_monom assms(2) diff_Suc_1 is_UP_monomE(1) is_UP_monomI monom_Suc(1) pderiv_closed pderiv_monom)
qed
qed
qed
qed
thus ?thesis using assms by blast
qed
lemma(in UP_cring) product_rule:
assumes "f ∈ carrier (UP R)"
assumes "g ∈ carrier (UP R)"
shows "pderiv (f ⊗⇘UP R⇙g) = (pderiv f ⊗⇘UP R⇙ g) ⊕⇘UP R⇙ (f ⊗⇘UP R⇙ pderiv g)"
proof(rule poly_induct3[of f])
show "f ∈ carrier P"
using assms unfolding P_def by blast
show "⋀p q. q ∈ carrier P ⟹
p ∈ carrier P ⟹
pderiv (p ⊗⇘UP R⇙ g) = pderiv p ⊗⇘UP R⇙ g ⊕⇘UP R⇙ p ⊗⇘UP R⇙ pderiv g ⟹
pderiv (q ⊗⇘UP R⇙ g) = pderiv q ⊗⇘UP R⇙ g ⊕⇘UP R⇙ q ⊗⇘UP R⇙ pderiv g ⟹
pderiv ((p ⊕⇘P⇙ q) ⊗⇘UP R⇙ g) = pderiv (p ⊕⇘P⇙ q) ⊗⇘UP R⇙ g ⊕⇘UP R⇙ (p ⊕⇘P⇙ q) ⊗⇘UP R⇙ pderiv g"
proof- fix p q
assume A: "q ∈ carrier P" "p ∈ carrier P"
"pderiv (p ⊗⇘UP R⇙ g) = pderiv p ⊗⇘UP R⇙ g ⊕⇘UP R⇙ p ⊗⇘UP R⇙ pderiv g"
"pderiv (q ⊗⇘UP R⇙ g) = pderiv q ⊗⇘UP R⇙ g ⊕⇘UP R⇙ q ⊗⇘UP R⇙ pderiv g"
have 0: "(p ⊕⇘P⇙ q) ⊗⇘UP R⇙ g = p ⊗⇘UP R⇙ g ⊕⇘UP R⇙ q ⊗⇘UP R⇙ g"
using A assms unfolding P_def using P_def UP_l_distr by blast
have 1: "pderiv ((p ⊕⇘P⇙ q) ⊗⇘UP R⇙ g) = pderiv (p ⊗⇘UP R⇙ g) ⊕⇘UP R⇙ pderiv (q ⊗⇘UP R⇙ g)"
unfolding 0 using pderiv_add[of "p ⊗⇘P⇙ g" "q ⊗⇘P⇙ g"] unfolding P_def
using A(1) A(2) P_def UP_mult_closed assms(2) by blast
have 2: "pderiv ((p ⊕⇘P⇙ q) ⊗⇘UP R⇙ g) = pderiv p ⊗⇘UP R⇙ g ⊕⇘UP R⇙ p ⊗⇘UP R⇙ pderiv g ⊕⇘UP R⇙ (pderiv q ⊗⇘UP R⇙ g ⊕⇘UP R⇙ q ⊗⇘UP R⇙ pderiv g)"
unfolding 1 A by blast
have 3: "pderiv ((p ⊕⇘P⇙ q) ⊗⇘UP R⇙ g) = pderiv p ⊗⇘UP R⇙ g ⊕⇘UP R⇙ pderiv q ⊗⇘UP R⇙ g ⊕⇘UP R⇙ p ⊗⇘UP R⇙ pderiv g ⊕⇘UP R⇙ q ⊗⇘UP R⇙ pderiv g"
using A assms by (metis "1" P.add.m_assoc P.add.m_lcomm P.m_closed P_def pderiv_closed)
have 4: "pderiv ((p ⊕⇘P⇙ q) ⊗⇘UP R⇙ g) = (pderiv p ⊗⇘UP R⇙ g ⊕⇘UP R⇙ pderiv q ⊗⇘UP R⇙ g) ⊕⇘UP R⇙ (p ⊗⇘UP R⇙ pderiv g ⊕⇘UP R⇙ q ⊗⇘UP R⇙ pderiv g)"
unfolding 3 using A assms P_def UP_a_assoc UP_a_closed UP_mult_closed pderiv_closed by auto
have 5: "pderiv ((p ⊕⇘P⇙ q) ⊗⇘UP R⇙ g) = ((pderiv p ⊕⇘UP R⇙ pderiv q) ⊗⇘UP R⇙ g) ⊕⇘UP R⇙ ((p ⊕⇘UP R⇙ q) ⊗⇘UP R⇙ pderiv g)"
unfolding 4 using A assms by (metis P.l_distr P_def pderiv_closed)
have 6: "pderiv ((p ⊕⇘P⇙ q) ⊗⇘UP R⇙ g) = ((pderiv (p ⊕⇘P⇙ q)) ⊗⇘UP R⇙ g) ⊕⇘UP R⇙ ((p ⊕⇘UP R⇙ q) ⊗⇘UP R⇙ pderiv g)"
unfolding 5 using A assms
by (metis P_def pderiv_add)
show "pderiv ((p ⊕⇘P⇙ q) ⊗⇘UP R⇙ g) = pderiv (p ⊕⇘P⇙ q) ⊗⇘UP R⇙ g ⊕⇘UP R⇙ (p ⊕⇘P⇙ q) ⊗⇘UP R⇙ pderiv g"
unfolding 6 using A assms P_def by blast
qed
show "⋀a n. a ∈ carrier R ⟹
pderiv (up_ring.monom P a n ⊗⇘UP R⇙ g) = pderiv (up_ring.monom P a n) ⊗⇘UP R⇙ g ⊕⇘UP R⇙ up_ring.monom P a n ⊗⇘UP R⇙ pderiv g"
using P_def UP_m_comm assms(2) is_UP_monomE(1) monom_is_UP_monom(1) monom_product_rule pderiv_closed by presburger
qed
subsection‹The Chain Rule›
lemma(in UP_cring) chain_rule:
assumes "f ∈ carrier P"
assumes "g ∈ carrier P"
shows "pderiv (compose R f g) = compose R (pderiv f) g ⊗⇘UP R⇙ pderiv g"
proof(rule poly_induct3[of f])
show "f ∈ carrier P"
using assms by blast
show "⋀p q. q ∈ carrier P ⟹
p ∈ carrier P ⟹
pderiv (Cring_Poly.compose R p g) = Cring_Poly.compose R (pderiv p) g ⊗⇘UP R⇙ pderiv g ⟹
pderiv (Cring_Poly.compose R q g) = Cring_Poly.compose R (pderiv q) g ⊗⇘UP R⇙ pderiv g ⟹
pderiv (Cring_Poly.compose R (p ⊕⇘P⇙ q) g) = Cring_Poly.compose R (pderiv (p ⊕⇘P⇙ q)) g ⊗⇘UP R⇙ pderiv g"
using pderiv_add sub_add
by (smt (verit) P_def UP_a_closed UP_m_comm UP_r_distr assms(2) pderiv_closed sub_closed)
show "⋀a n. a ∈ carrier R ⟹
pderiv (compose R (up_ring.monom P a n) g) = compose R (pderiv (up_ring.monom P a n)) g ⊗⇘UP R⇙ pderiv g"
proof-
fix a n assume A: "a ∈ carrier R"
show "pderiv (compose R (up_ring.monom P a n) g) = compose R (pderiv (up_ring.monom P a n)) g ⊗⇘UP R⇙ pderiv g"
proof(induction n)
case 0
have 00: "(compose R (up_ring.monom P a 0) g) = (up_ring.monom P a 0)"
using A P_def assms(2) deg_const is_UP_monom_def monom_is_UP_monom(1) sub_const by presburger
have 01: "pderiv (up_ring.monom P a 0) = 𝟬⇘P⇙"
using A pderiv_const by blast
show ?case unfolding 00 01
by (metis P.l_null P_def UP_zero_closed assms(2) deg_zero pderiv_closed sub_const)
next
case (Suc n)
show "pderiv (Cring_Poly.compose R (up_ring.monom P a (Suc n)) g) = Cring_Poly.compose R (pderiv (up_ring.monom P a (Suc n))) g ⊗⇘UP R⇙ pderiv g"
proof(cases "n = 0")
case True
have 0: "compose R (up_ring.monom P a (Suc n)) g = a ⊙⇘P⇙ g"
using A assms sub_monom_deg_one[of g a] unfolding True using One_nat_def
by presburger
have 1: "(pderiv (up_ring.monom P a (Suc n))) = up_ring.monom P a 0"
unfolding True
proof -
have "pderiv (up_ring.monom P a 0) = 𝟬⇘P⇙"
using A pderiv_const by blast
then show "pderiv (up_ring.monom P a (Suc 0)) = up_ring.monom P a 0"
using A lcf_monom(1) P_def X_closed deg_const deg_nzero_nzero is_UP_monomE(1) monom_Suc(2) monom_is_UP_monom(1) monom_rep_X_pow pderiv_monom poly_shift_degree_zero poly_shift_eq sub_monom(2) sub_monom_deg_one to_poly_inverse to_poly_mult_simp(2)
by (metis (no_types, lifting) P.l_null P.r_zero X_poly_def times_x_product_rule)
qed
then show ?thesis unfolding 0 1
using A P_def assms(2) deg_const is_UP_monomE(1) monom_is_UP_monom(1) monom_mult_is_smult pderiv_closed pderiv_smult sub_const
by presburger
next
case False
have 0: "compose R (up_ring.monom P a (Suc n)) g = (compose R (up_ring.monom P a n) g) ⊗⇘P⇙ (compose R (up_ring.monom P 𝟭 1) g)"
using assms A by (metis R.one_closed monom_Suc(2) monom_closed sub_mult)
have 1: "compose R (up_ring.monom P a (Suc n)) g = (compose R (up_ring.monom P a n) g) ⊗⇘P⇙ g"
unfolding 0 using A assms
by (metis P_def R.one_closed UP_cring.lcf_monom(1) UP_cring.to_poly_inverse UP_cring_axioms UP_l_one UP_one_closed deg_one monom_one sub_monom_deg_one to_poly_mult_simp(1))
have 2: "pderiv (compose R (up_ring.monom P a (Suc n)) g ) =
((pderiv (compose R (up_ring.monom P a n) g)) ⊗⇘P⇙ g) ⊕⇘P⇙ ((compose R (up_ring.monom P a n) g) ⊗⇘P⇙ pderiv g)"
unfolding 1 unfolding P_def apply(rule product_rule)
using A assms unfolding P_def using P_def is_UP_monomE(1) is_UP_monomI rev_sub_closed sub_rev_sub apply presburger
using assms unfolding P_def by blast
have 3: "pderiv (compose R (up_ring.monom P a (Suc n)) g ) =
(compose R (pderiv (up_ring.monom P a n)) g ⊗⇘UP R⇙ pderiv g ⊗⇘P⇙ g) ⊕⇘P⇙ ((compose R (up_ring.monom P a n) g) ⊗⇘P⇙ pderiv g)"
unfolding 2 Suc by blast
have 4: "pderiv (compose R (up_ring.monom P a (Suc n)) g ) =
((compose R (pderiv (up_ring.monom P a n)) g ⊗⇘P⇙ g) ⊗⇘UP R⇙ pderiv g) ⊕⇘P⇙ ((compose R (up_ring.monom P a n) g) ⊗⇘P⇙ pderiv g)"
unfolding 3 using A assms m_assoc m_comm
by (smt (verit) P_def monom_closed monom_rep_X_pow pderiv_closed sub_closed)
have 5: "pderiv (compose R (up_ring.monom P a (Suc n)) g ) =
((compose R (pderiv (up_ring.monom P a n)) g ⊗⇘P⇙ g) ⊕⇘P⇙ (compose R (up_ring.monom P a n) g)) ⊗⇘P⇙ pderiv g"
unfolding 4 using A assms
by (metis P.l_distr P.m_closed P_def UP_cring.pderiv_closed UP_cring_axioms monom_closed sub_closed)
have 6: "compose R (pderiv (up_ring.monom P a n)) g ⊗⇘P⇙ g = [n]⋅⇘P⇙compose R ((up_ring.monom P a n)) g"
proof-
have 60: "(pderiv (up_ring.monom P a n)) = (up_ring.monom P ([n]⋅a) (n-1))"
using A assms pderiv_monom by blast
have 61: "compose R (pderiv (up_ring.monom P a n)) g ⊗⇘P⇙ g = compose R ((up_ring.monom P ([n]⋅a) (n-1))) g ⊗⇘P⇙ (compose R (up_ring.monom P 𝟭 1) g)"
unfolding 60 using A assms sub_monom_deg_one[of g 𝟭 ] R.one_closed smult_one by presburger
have 62: "compose R (pderiv (up_ring.monom P a n)) g ⊗⇘P⇙ g = compose R (up_ring.monom P ([n]⋅a) n) g"
unfolding 61 using False A assms sub_mult[of g "up_ring.monom P ([n] ⋅ a) (n - 1)" "up_ring.monom P 𝟭 1" ] monom_mult[of "[n]⋅a" 𝟭 "n-1" 1]
by (metis Nat.add_0_right R.add.nat_pow_closed R.one_closed R.r_one Suc_eq_plus1 add_eq_if monom_closed)
have 63: "⋀k::nat. Cring_Poly.compose R (up_ring.monom P ([k] ⋅ a) n) g = [k] ⋅⇘P⇙Cring_Poly.compose R (up_ring.monom P a n) g"
proof- fix k::nat show "Cring_Poly.compose R (up_ring.monom P ([k] ⋅ a) n) g = [k] ⋅⇘P⇙Cring_Poly.compose R (up_ring.monom P a n) g"
apply(induction k)
using UP_zero_closed assms(2) deg_zero monom_zero sub_const
apply (metis A P.add.nat_pow_0 add_nat_pow_monom)
proof-
fix k::nat
assume a: "Cring_Poly.compose R (monom P ([k] ⋅ a) n) g =
[k] ⋅⇘P⇙ Cring_Poly.compose R (monom P a n) g"
have 0: "(monom P ([Suc k] ⋅ a) n) = [Suc k] ⋅ a ⊙⇘P⇙(monom P 𝟭 n)"
by (simp add: A monic_monom_smult)
have 1: "(monom P ([Suc k] ⋅ a) n) = [k] ⋅ a ⊙⇘P⇙(monom P 𝟭 n) ⊕⇘P⇙a ⊙⇘P⇙(monom P 𝟭 n) "
unfolding 0
by (simp add: A UP_smult_l_distr)
show "Cring_Poly.compose R (monom P ([Suc k] ⋅ a) n) g =
[Suc k] ⋅⇘P⇙ (Cring_Poly.compose R (monom P a n) g) "
unfolding 1
by (simp add: A a assms(2) monic_monom_smult sub_add)
qed
qed
have 64: "Cring_Poly.compose R (up_ring.monom P ([n] ⋅ a) n) g = [n] ⋅⇘P⇙Cring_Poly.compose R (up_ring.monom P a n) g"
using 63 by blast
show ?thesis unfolding 62 64 by blast
qed
have 63: "⋀k::nat. Cring_Poly.compose R (up_ring.monom P ([k] ⋅ a) n) g = [k] ⋅⇘P⇙Cring_Poly.compose R (up_ring.monom P a n) g"
proof- fix k::nat show "Cring_Poly.compose R (up_ring.monom P ([k] ⋅ a) n) g = [k] ⋅⇘P⇙Cring_Poly.compose R (up_ring.monom P a n) g"
apply(induction k)
using UP_zero_closed assms(2) deg_zero monom_zero sub_const
apply (metis A P.add.nat_pow_0 add_nat_pow_monom)
using A P.add.nat_pow_Suc add_nat_pow_monom assms(2) is_UP_monomE(1) monom_is_UP_monom(1) rev_sub_add sub_rev_sub
by (metis P.add.nat_pow_closed)
qed
have 7: "([n] ⋅⇘P⇙ Cring_Poly.compose R (up_ring.monom P a n) g ⊕⇘P⇙ Cring_Poly.compose R (up_ring.monom P a n) g) =
[Suc n] ⋅⇘P⇙ (Cring_Poly.compose R (up_ring.monom P a n) g)"
using A assms P.add.nat_pow_Suc by presburger
have 8: "[Suc n] ⋅⇘P⇙ Cring_Poly.compose R (up_ring.monom P a n) g ⊗⇘P⇙ pderiv g = Cring_Poly.compose R (up_ring.monom P ([Suc n] ⋅ a) n) g ⊗⇘P⇙ pderiv g"
unfolding 63[of "Suc n"] by blast
show ?thesis unfolding 5 6 7 8 using A assms pderiv_monom[of "a" "Suc n"]
using P_def diff_Suc_1 by metis
qed
qed
qed
qed
lemma deriv_prod_rule_times_monom:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
assumes "q ∈ carrier P"
shows "deriv ((monom P a n) ⊗⇘P⇙ q) b = (deriv (monom P a n) b) ⊗ (to_fun q b) ⊕ (to_fun (monom P a n) b) ⊗ deriv q b"
proof(rule poly_induct3[of q])
show "q ∈ carrier P"
using assms by simp
show " ⋀p q. q ∈ carrier P ⟹
p ∈ carrier P ⟹
deriv (monom P a n ⊗⇘P⇙ p) b = deriv (monom P a n) b ⊗ to_fun p b ⊕ to_fun (monom P a n) b ⊗ deriv p b ⟹
deriv (monom P a n ⊗⇘P⇙ q) b = deriv (monom P a n) b ⊗ to_fun q b ⊕ to_fun (monom P a n) b ⊗ deriv q b ⟹
deriv (monom P a n ⊗⇘P⇙ (p ⊕⇘P⇙ q)) b = deriv (monom P a n) b ⊗ to_fun (p ⊕⇘P⇙ q) b ⊕ to_fun (monom P a n) b ⊗ deriv (p ⊕⇘P⇙ q) b"
proof- fix p q assume A: "q ∈ carrier P" " p ∈ carrier P"
"deriv (monom P a n ⊗⇘P⇙ p) b = deriv (monom P a n) b ⊗ to_fun p b ⊕ to_fun (monom P a n) b ⊗ deriv p b"
"deriv (monom P a n ⊗⇘P⇙ q) b = deriv (monom P a n) b ⊗ to_fun q b ⊕ to_fun (monom P a n) b ⊗ deriv q b"
have "deriv (monom P a n ⊗⇘P⇙ (p ⊕⇘P⇙ q)) b = deriv (monom P a n) b ⊗ to_fun p b ⊕ to_fun (monom P a n) b ⊗ deriv p b
⊕deriv (monom P a n) b ⊗ to_fun q b ⊕ to_fun (monom P a n) b ⊗ deriv q b"
using A assms
by (simp add: P.r_distr R.add.m_assoc deriv_add deriv_closed to_fun_closed)
hence "deriv (monom P a n ⊗⇘P⇙ (p ⊕⇘P⇙ q)) b = deriv (monom P a n) b ⊗ to_fun p b ⊕deriv (monom P a n) b ⊗ to_fun q b
⊕ to_fun (monom P a n) b ⊗ deriv p b ⊕ to_fun (monom P a n) b ⊗ deriv q b"
using A(1) A(2) R.add.m_assoc R.add.m_comm assms(1) assms(2) deriv_closed to_fun_closed by auto
hence "deriv (monom P a n ⊗⇘P⇙ (p ⊕⇘P⇙ q)) b = deriv (monom P a n) b ⊗ (to_fun p b ⊕ to_fun q b)
⊕ to_fun (monom P a n) b ⊗ (deriv p b ⊕ deriv q b)"
by (simp add: A(1) A(2) R.add.m_assoc R.r_distr assms(1) assms(2) deriv_closed to_fun_closed)
thus "deriv (monom P a n ⊗⇘P⇙ (p ⊕⇘P⇙ q)) b = deriv (monom P a n) b ⊗ to_fun (p ⊕⇘P⇙ q) b ⊕ to_fun (monom P a n) b ⊗ deriv (p ⊕⇘P⇙ q) b"
by (simp add: A(1) A(2) assms(2) deriv_add to_fun_plus)
qed
show "⋀c m. c ∈ carrier R ⟹ deriv (monom P a n ⊗⇘P⇙ monom P c m) b =
deriv (monom P a n) b ⊗ to_fun (monom P c m) b
⊕ to_fun (monom P a n) b ⊗ deriv (monom P c m) b"
proof- fix c m assume A: "c ∈ carrier R"
show "deriv (monom P a n ⊗⇘P⇙ monom P c m) b = deriv (monom P a n) b ⊗ to_fun (monom P c m) b ⊕ to_fun (monom P a n) b ⊗ deriv (monom P c m) b"
proof(cases "n = 0")
case True
have LHS: "deriv (monom P a n ⊗⇘P⇙ monom P c m) b = deriv (monom P (a ⊗ c) m) b"
by (metis A True add.left_neutral assms(1) monom_mult)
have RHS: "deriv (monom P a n) b ⊗ to_fun (monom P c m) b ⊕ to_fun (monom P a n) b ⊗ deriv (monom P c m) b
= a ⊗ deriv (monom P c m) b "
using deriv_const to_fun_monom A True assms(1) assms(2) deriv_closed by auto
show ?thesis using A assms LHS RHS deriv_monom
by (smt (verit) R.add.nat_pow_closed R.add_pow_rdistr R.m_assoc R.m_closed R.nat_pow_closed)
next
case False
show ?thesis
proof(cases "m = 0")
case True
have LHS: "deriv (monom P a n ⊗⇘P⇙ monom P c m) b = deriv (monom P (a ⊗ c) n) b"
by (metis A True add.comm_neutral assms(1) monom_mult)
have RHS: "deriv (monom P a n) b ⊗ to_fun (monom P c m) b ⊕ to_fun (monom P a n) b ⊗ deriv (monom P c m) b
= c ⊗ deriv (monom P a n) b "
by (metis (no_types, lifting) A lcf_monom(1) P_def R.m_closed R.m_comm R.r_null
R.r_zero True UP_cring.to_fun_ctrm UP_cring_axioms assms(1) assms(2) deg_const
deriv_closed deriv_const to_fun_closed monom_closed)
show ?thesis using LHS RHS deriv_monom A assms
by (smt (verit) R.add.nat_pow_closed R.add_pow_ldistr R.m_assoc R.m_closed R.m_comm R.nat_pow_closed)
next
case F: False
have pos: "n > 0" "m >0"
using F False by auto
have RHS: "deriv (monom P a n ⊗⇘P⇙ monom P c m) b = [(n + m)] ⋅ (a ⊗ c) ⊗ b [^] (n + m - 1)"
using deriv_monom[of "a ⊗ c" b "n + m"] monom_mult[of a c n m]
by (simp add: A assms(1) assms(2))
have LHS: "deriv (monom P a n) b ⊗ to_fun (monom P c m) b ⊕ to_fun (monom P a n) b ⊗ deriv (monom P c m) b
= [n]⋅a ⊗(b[^](n-1)) ⊗ c ⊗ b[^]m ⊕ a ⊗ b[^]n ⊗ [m]⋅c ⊗(b[^](m-1))"
using deriv_monom[of a b n] to_fun_monom[of a b n]
deriv_monom[of c b m] to_fun_monom[of c b m] A assms
by (simp add: R.m_assoc)
have 0: "[n]⋅a ⊗ (b[^](n-1)) ⊗ c ⊗ b[^]m = [n]⋅a ⊗ c ⊗ b[^](n + m -1) "
proof-
have "[n]⋅a ⊗ (b[^](n-1)) ⊗ c ⊗ b[^]m = [n]⋅a ⊗ c ⊗ (b[^](n-1)) ⊗ b[^]m"
by (simp add: A R.m_lcomm R.semiring_axioms assms(1) assms(2) semiring.semiring_simprules(8))
hence "[n]⋅a ⊗ (b[^](n-1)) ⊗ c ⊗ b[^]m = [n]⋅a ⊗ c ⊗ ((b[^](n-1)) ⊗ b[^]m)"
by (simp add: A R.m_assoc assms(1) assms(2))
thus ?thesis
by (simp add: False R.nat_pow_mult add_eq_if assms(2))
qed
have 1: "a ⊗ b[^]n ⊗ [m]⋅c ⊗(b[^](m-1)) = a ⊗ [m]⋅c ⊗ b[^](n + m -1)"
proof-
have "a ⊗ b[^]n ⊗ [m]⋅c ⊗(b[^](m-1)) = a ⊗ [m]⋅c ⊗ b[^]n ⊗(b[^](m-1))"
using A R.m_comm R.m_lcomm assms(1) assms(2) by auto
hence "a ⊗ b[^]n ⊗ [m]⋅c ⊗(b[^](m-1)) = a ⊗ [m]⋅c ⊗ (b[^]n ⊗(b[^](m-1)))"
by (simp add: A R.m_assoc assms(1) assms(2))
thus ?thesis
by (simp add: F R.nat_pow_mult add.commute add_eq_if assms(2))
qed
have LHS: "deriv (monom P a n) b ⊗ to_fun (monom P c m) b ⊕ to_fun (monom P a n) b ⊗ deriv (monom P c m) b
= [n]⋅a ⊗ c ⊗ b[^](n + m -1) ⊕ a ⊗ [m]⋅c ⊗ b[^](n + m -1)"
using LHS 0 1
by simp
hence LHS: "deriv (monom P a n) b ⊗ to_fun (monom P c m) b ⊕ to_fun (monom P a n) b ⊗ deriv (monom P c m) b
= [n]⋅ (a ⊗ c ⊗ b[^](n + m -1)) ⊕ [m]⋅ (a ⊗ c ⊗ b[^](n + m -1))"
by (simp add: A R.add_pow_ldistr R.add_pow_rdistr assms(1) assms(2))
show ?thesis using LHS RHS
by (simp add: A R.add.nat_pow_mult R.add_pow_ldistr assms(1) assms(2))
qed
qed
qed
qed
lemma deriv_prod_rule:
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
assumes "a ∈ carrier R"
shows "deriv (p ⊗⇘P⇙ q) a = deriv p a ⊗ (to_fun q a) ⊕ (to_fun p a) ⊗ deriv q a"
proof(rule poly_induct3[of p])
show "p ∈ carrier P"
using assms(1) by simp
show " ⋀p qa.
qa ∈ carrier P ⟹
p ∈ carrier P ⟹
deriv (p ⊗⇘P⇙ q) a = deriv p a ⊗ to_fun q a ⊕ to_fun p a ⊗ deriv q a ⟹
deriv (qa ⊗⇘P⇙ q) a = deriv qa a ⊗ to_fun q a ⊕ to_fun qa a ⊗ deriv q a ⟹
deriv ((p ⊕⇘P⇙ qa) ⊗⇘P⇙ q) a = deriv (p ⊕⇘P⇙ qa) a ⊗ to_fun q a ⊕ to_fun (p ⊕⇘P⇙ qa) a ⊗ deriv q a"
proof- fix f g assume A: "f ∈ carrier P" "g ∈ carrier P"
"deriv (f ⊗⇘P⇙ q) a = deriv f a ⊗ to_fun q a ⊕ to_fun f a ⊗ deriv q a"
"deriv (g ⊗⇘P⇙ q) a = deriv g a ⊗ to_fun q a ⊕ to_fun g a ⊗ deriv q a"
have "deriv ((f ⊕⇘P⇙ g) ⊗⇘P⇙ q) a = deriv f a ⊗ to_fun q a ⊕ to_fun f a ⊗ deriv q a ⊕
deriv g a ⊗ to_fun q a ⊕ to_fun g a ⊗ deriv q a"
using A deriv_add
by (simp add: P.l_distr R.add.m_assoc assms(2) assms(3) deriv_closed to_fun_closed)
hence "deriv ((f ⊕⇘P⇙ g) ⊗⇘P⇙ q) a = deriv f a ⊗ to_fun q a ⊕ deriv g a ⊗ to_fun q a ⊕
to_fun f a ⊗ deriv q a ⊕ to_fun g a ⊗ deriv q a"
using R.a_comm R.a_assoc deriv_closed to_fun_closed assms
by (simp add: A(1) A(2))
hence "deriv ((f ⊕⇘P⇙ g) ⊗⇘P⇙ q) a = (deriv f a ⊗ to_fun q a ⊕ deriv g a ⊗ to_fun q a) ⊕
(to_fun f a ⊗ deriv q a ⊕ to_fun g a ⊗ deriv q a)"
by (simp add: A(1) A(2) R.add.m_assoc assms(2) assms(3) deriv_closed to_fun_closed)
thus "deriv ((f ⊕⇘P⇙ g) ⊗⇘P⇙ q) a = deriv (f ⊕⇘P⇙ g) a ⊗ to_fun q a ⊕ to_fun (f ⊕⇘P⇙ g) a ⊗ deriv q a"
by (simp add: A(1) A(2) R.l_distr assms(2) assms(3) deriv_add deriv_closed to_fun_closed to_fun_plus)
qed
show "⋀aa n. aa ∈ carrier R ⟹ deriv (monom P aa n ⊗⇘P⇙ q) a = deriv (monom P aa n) a ⊗ to_fun q a ⊕ to_fun (monom P aa n) a ⊗ deriv q a"
using deriv_prod_rule_times_monom
by (simp add: assms(2) assms(3))
qed
lemma pderiv_eval_deriv_monom:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "to_fun (pderiv (monom P a n)) b = deriv (monom P a n) b"
using deriv_monom assms pderiv_monom
by (simp add: P_def UP_cring.to_fun_monom UP_cring_axioms)
lemma pderiv_eval_deriv:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "deriv f a = to_fun (pderiv f) a"
apply(rule poly_induct3[of f])
apply (simp add: assms(1))
using assms(2) deriv_add to_fun_plus pderiv_add pderiv_closed apply presburger
using assms(2) pderiv_eval_deriv_monom
by presburger
text‹Taking taylor expansions commutes with taking derivatives:›
lemma(in UP_cring) taylor_expansion_pderiv_comm:
assumes "f ∈ carrier (UP R)"
assumes "c ∈ carrier R"
shows "pderiv (taylor_expansion R c f) = taylor_expansion R c (pderiv f)"
apply(rule poly_induct3[of f])
using assms unfolding P_def apply blast
proof-
fix p q assume A: " q ∈ carrier (UP R)" "p ∈ carrier (UP R)"
"pderiv (taylor_expansion R c p) = taylor_expansion R c (pderiv p)"
"pderiv (taylor_expansion R c q) = taylor_expansion R c (pderiv q)"
have 0: " pderiv (taylor_expansion R c (p ⊕⇘UP R⇙ q)) = pderiv (taylor_expansion R c p ⊕⇘UP R⇙ taylor_expansion R c q)"
using A P_def taylor_expansion_add assms(2) by presburger
show "pderiv (taylor_expansion R c (p ⊕⇘UP R⇙ q)) = taylor_expansion R c (pderiv (p ⊕⇘UP R⇙ q))"
unfolding 0
using A(1) A(2) A(3) A(4) taylor_def UP_cring.taylor_closed UP_cring.taylor_expansion_add UP_cring.pderiv_add UP_cring.pderiv_closed UP_cring_axioms assms(2) by fastforce
next
fix a n assume A: "a ∈ carrier R"
show "pderiv (taylor_expansion R c (up_ring.monom (UP R) a n)) = taylor_expansion R c (pderiv (up_ring.monom (UP R) a n))"
proof(cases "n = 0")
case True
have 0: "deg R (taylor_expansion R c (up_ring.monom (UP R) a n)) = 0"
unfolding True
using P_def A assms taylor_def taylor_deg deg_const is_UP_monomE(1) monom_is_UP_monom(2) by presburger
have 1: "(pderiv (up_ring.monom (UP R) a n)) = 𝟬⇘P⇙"
unfolding True using P_def A assms pderiv_const by blast
show ?thesis unfolding 1 using 0 A assms P_def
by (metis P.add.right_cancel taylor_closed taylor_def taylor_expansion_add UP_l_zero UP_zero_closed monom_closed pderiv_deg_0)
next
case False
have 0: "pderiv (up_ring.monom (UP R) a n) = (up_ring.monom (UP R) ([n]⋅a) (n-1))"
using A
by (simp add: UP_cring.pderiv_monom UP_cring_axioms)
have 1: "pderiv (taylor_expansion R c (up_ring.monom (UP R) a n)) = (Cring_Poly.compose R (up_ring.monom (UP R) ([n]⋅a) (n-1)) (X_plus c)) ⊗⇘P⇙ pderiv (X_plus c)"
using chain_rule[of "up_ring.monom (UP R) a n" "X_plus c"] unfolding 0 taylor_expansion_def
using A P_def X_plus_closed assms(2) is_UP_monom_def monom_is_UP_monom(1) by presburger
have 2: "pderiv (X_plus c) = 𝟭⇘P⇙"
using pderiv_add[of "X_poly R" "to_poly c"] P.l_null P.l_one P.r_zero P_def R.one_closed X_closed
X_poly_def X_poly_plus_def assms(2) monom_one pderiv_const to_poly_closed to_polynomial_def
by (metis times_x_product_rule)
show ?thesis unfolding 1 0 2 taylor_expansion_def
by (metis "1" "2" A P.l_one P_def R.add.nat_pow_closed UP_m_comm UP_one_closed X_plus_closed assms(2) monom_closed sub_closed taylor_expansion_def)
qed
qed
subsection‹Linear Substitutions›
lemma(in UP_ring) lcoeff_Lcf:
assumes "f ∈ carrier P"
shows "lcoeff f = lcf f"
unfolding P_def
using assms coeff_simp[of f] by metis
lemma(in UP_cring) linear_sub_cfs:
assumes "f ∈ carrier (UP R)"
assumes "d ∈ carrier R"
assumes "g = compose R f (up_ring.monom (UP R) d 1)"
shows "g i = d[^]i ⊗ f i"
proof-
have 0: "(up_ring.monom (UP R) d 1) ∈ carrier (UP R)"
using assms by (meson R.ring_axioms UP_ring.intro UP_ring.monom_closed)
have 1: "(∀i. compose R f (up_ring.monom (UP R) d 1) i = d[^]i ⊗ f i)"
apply(rule poly_induct3[of f])
using assms unfolding P_def apply blast
proof-
show "⋀p q. q ∈ carrier (UP R) ⟹
p ∈ carrier (UP R) ⟹
∀i. Cring_Poly.compose R p (up_ring.monom (UP R) d 1) i = d [^] i ⊗ p i ⟹
∀i. Cring_Poly.compose R q (up_ring.monom (UP R) d 1) i = d [^] i ⊗ q i ⟹
∀i. Cring_Poly.compose R (p ⊕⇘UP R⇙ q) (up_ring.monom (UP R) d 1) i = d [^] i ⊗ (p ⊕⇘UP R⇙ q) i"
proof
fix p q i
assume A: "q ∈ carrier (UP R)"
"p ∈ carrier (UP R)"
"∀i. Cring_Poly.compose R p (up_ring.monom (UP R) d 1) i = d [^] i ⊗ p i"
"∀i. Cring_Poly.compose R q (up_ring.monom (UP R) d 1) i = d [^] i ⊗ q i"
show "Cring_Poly.compose R (p ⊕⇘UP R⇙ q) (up_ring.monom (UP R) d 1) i = d [^] i ⊗ (p ⊕⇘UP R⇙ q) i"
proof-
have 1: "Cring_Poly.compose R (p ⊕⇘UP R⇙ q) (up_ring.monom (UP R) d 1) =
Cring_Poly.compose R p (up_ring.monom (UP R) d 1) ⊕⇘UP R⇙ Cring_Poly.compose R q (up_ring.monom (UP R) d 1)"
using A(1) A(2) sub_add[of "up_ring.monom (UP R) d 1" q p] unfolding P_def
using "0" P_def sub_add by blast
have 2: "Cring_Poly.compose R (p ⊕⇘UP R⇙ q) (up_ring.monom (UP R) d 1) i =
Cring_Poly.compose R p (up_ring.monom (UP R) d 1) i ⊕ Cring_Poly.compose R q (up_ring.monom (UP R) d 1) i"
using 1 by (metis (no_types, lifting) "0" A(1) A(2) P_def cfs_add sub_closed)
have 3: "Cring_Poly.compose R (p ⊕⇘UP R⇙ q) (up_ring.monom (UP R) d 1) i = d [^] i ⊗ p i ⊕ d [^] i ⊗ q i"
unfolding 2 using A by presburger
have 4: "Cring_Poly.compose R (p ⊕⇘UP R⇙ q) (up_ring.monom (UP R) d 1) i = d [^] i ⊗ (p i ⊕ q i)"
using "3" A(1) A(2) R.nat_pow_closed R.r_distr UP_car_memE(1) assms(2) by presburger
thus "Cring_Poly.compose R (p ⊕⇘UP R⇙ q) (up_ring.monom (UP R) d 1) i = d [^] i ⊗ (p ⊕⇘UP R⇙ q) i"
unfolding 4 using A(1) A(2) P_def cfs_add by presburger
qed
qed
show "⋀a n. a ∈ carrier R ⟹
∀i. Cring_Poly.compose R (up_ring.monom (UP R) a n) (up_ring.monom (UP R) d 1) i = d [^] i ⊗ up_ring.monom (UP R) a n i"
proof fix a n i assume A: "a ∈ carrier R"
have 0: "Cring_Poly.compose R (up_ring.monom (UP R) a n) (up_ring.monom (UP R) d 1) =
a ⊙⇘UP R⇙(up_ring.monom (UP R) d 1)[^]⇘UP R⇙n"
using assms A 0 P_def monom_sub by blast
have 1: "Cring_Poly.compose R (up_ring.monom (UP R) a n) (up_ring.monom (UP R) d 1) =
a ⊙⇘UP R⇙ (d[^]n ⊙⇘UP R⇙(up_ring.monom (UP R) 𝟭 n))"
unfolding 0 using A assms
by (metis P_def R.nat_pow_closed monic_monom_smult monom_pow mult.left_neutral)
have 2: "Cring_Poly.compose R (up_ring.monom (UP R) a n) (up_ring.monom (UP R) d 1) =
(a ⊗d[^]n)⊙⇘UP R⇙(up_ring.monom (UP R) 𝟭 n)"
unfolding 1 using A assms
by (metis R.nat_pow_closed R.one_closed R.ring_axioms UP_ring.UP_smult_assoc1 UP_ring.intro UP_ring.monom_closed)
show "Cring_Poly.compose R (up_ring.monom (UP R) a n) (up_ring.monom (UP R) d 1) i = d [^] i ⊗ up_ring.monom (UP R) a n i"
apply(cases "i = n")
unfolding 2 using A P_def R.m_closed R.m_comm R.nat_pow_closed assms(2) cfs_monom monic_monom_smult apply presburger
using A P_def R.m_closed R.nat_pow_closed R.r_null assms(2) cfs_monom monic_monom_smult by presburger
qed
qed
show ?thesis using 1 unfolding assms
by blast
qed
lemma(in UP_cring) linear_sub_deriv:
assumes "f ∈ carrier (UP R)"
assumes "d ∈ carrier R"
assumes "g = compose R f (up_ring.monom (UP R) d 1)"
assumes "c ∈ carrier R"
shows "pderiv g = d ⊙⇘UP R⇙ compose R (pderiv f) (up_ring.monom (UP R) d 1)"
unfolding assms
proof(rule poly_induct3[of f])
show "f ∈ carrier P"
using assms unfolding P_def by blast
show "⋀ p q. q ∈ carrier P ⟹
p ∈ carrier P ⟹
pderiv (Cring_Poly.compose R p (up_ring.monom (UP R) d 1)) = d ⊙⇘UP R⇙ Cring_Poly.compose R (pderiv p) (up_ring.monom (UP R) d 1) ⟹
pderiv (Cring_Poly.compose R q (up_ring.monom (UP R) d 1)) = d ⊙⇘UP R⇙ Cring_Poly.compose R (pderiv q) (up_ring.monom (UP R) d 1) ⟹
pderiv (Cring_Poly.compose R (p ⊕⇘P⇙ q) (up_ring.monom (UP R) d 1)) =
d ⊙⇘UP R⇙ Cring_Poly.compose R (pderiv (p ⊕⇘P⇙ q)) (up_ring.monom (UP R) d 1)"
proof- fix p q assume A: "q ∈ carrier P" "p ∈ carrier P"
"pderiv (Cring_Poly.compose R p (up_ring.monom (UP R) d 1)) = d ⊙⇘UP R⇙ Cring_Poly.compose R (pderiv p) (up_ring.monom (UP R) d 1)"
"pderiv (Cring_Poly.compose R q (up_ring.monom (UP R) d 1)) = d ⊙⇘UP R⇙ Cring_Poly.compose R (pderiv q) (up_ring.monom (UP R) d 1)"
show " pderiv (Cring_Poly.compose R (p ⊕⇘P⇙ q) (up_ring.monom (UP R) d 1)) =
d ⊙⇘UP R⇙ Cring_Poly.compose R (pderiv (p ⊕⇘P⇙ q)) (up_ring.monom (UP R) d 1)"
using A assms P_def monom_closed pderiv_add pderiv_closed smult_r_distr sub_add sub_closed by force
qed
show "⋀a n. a ∈ carrier R ⟹
pderiv (Cring_Poly.compose R (up_ring.monom P a n) (up_ring.monom (UP R) d 1)) =
d ⊙⇘UP R⇙ Cring_Poly.compose R (pderiv (up_ring.monom P a n)) (up_ring.monom (UP R) d 1)"
proof- fix a n assume A: "a ∈ carrier R"
have "(Cring_Poly.compose R (up_ring.monom P a n) (up_ring.monom (UP R) d 1)) = a ⊙⇘UP R⇙ (up_ring.monom P d 1)[^]⇘UP R⇙ n"
using A assms sub_monom(2) P_def is_UP_monomE(1) monom_is_UP_monom(1) by blast
hence 0: "(Cring_Poly.compose R (up_ring.monom P a n) (up_ring.monom (UP R) d 1)) = a ⊙⇘UP R⇙ (up_ring.monom P (d[^]n) n)"
using A assms P_def monom_pow nat_mult_1 by metis
show "pderiv (Cring_Poly.compose R (up_ring.monom P a n) (up_ring.monom (UP R) d 1)) =
d ⊙⇘UP R⇙ Cring_Poly.compose R (pderiv (up_ring.monom P a n)) (up_ring.monom (UP R) d 1)"
proof(cases "n = 0")
case True
have T0: "pderiv (up_ring.monom P a n) = 𝟬⇘ UP R⇙" unfolding True
using A P_def pderiv_const by blast
have T1: "(Cring_Poly.compose R (up_ring.monom P a n) (up_ring.monom (UP R) d 1)) = up_ring.monom P a n"
unfolding True
using A assms P_def deg_const is_UP_monomE(1) monom_is_UP_monom(1) sub_const by presburger
thus ?thesis unfolding T0 T1
by (metis P.nat_pow_eone P_def UP_smult_closed UP_zero_closed X_closed assms(2) deg_zero monom_rep_X_pow smult_r_null sub_const)
next
case False
have F0: "pderiv (Cring_Poly.compose R (up_ring.monom P a n) (up_ring.monom (UP R) d 1)) = (a ⊙⇘UP R⇙ (up_ring.monom P ([n]⋅⇘R⇙(d[^]n))(n-1)))"
using A assms pderiv_monom unfolding 0
using P_def R.nat_pow_closed is_UP_monomE(1) monom_is_UP_monom(1) pderiv_smult by metis
have F1: "(pderiv (up_ring.monom P a n)) = up_ring.monom P ([n] ⋅ a) (n - 1)"
using A assms pderiv_monom[of a n] by blast
hence F2: "(pderiv (up_ring.monom P a n)) = ([n] ⋅ a) ⊙⇘UP R⇙up_ring.monom P 𝟭 (n - 1)"
using A P_def monic_monom_smult by auto
have F3: "Cring_Poly.compose R ((([n] ⋅ a) ⊙⇘UP R⇙ (up_ring.monom P 𝟭 (n - 1)))) (up_ring.monom (UP R) d 1) =
([n] ⋅ a) ⊙⇘UP R⇙ ((up_ring.monom (UP R) d 1)[^]⇘UP R⇙(n-1))"
using A F1 F2 P_def assms(2) monom_closed sub_monom(2) by fastforce
have F4: "Cring_Poly.compose R ((([n] ⋅ a) ⊙⇘UP R⇙ (up_ring.monom P 𝟭 (n - 1)))) (up_ring.monom (UP R) d 1) =
([n] ⋅ a) ⊙⇘UP R⇙ ((up_ring.monom (UP R) (d[^](n-1)) (n-1)))"
by (metis F3 P_def assms(2) monom_pow nat_mult_1)
have F5: "d ⊙⇘UP R⇙ (Cring_Poly.compose R (pderiv (up_ring.monom P a n)) (up_ring.monom (UP R) d 1)) =
(d ⊗([n] ⋅ a)) ⊙⇘UP R⇙ up_ring.monom (UP R) (d [^] (n - 1)) (n - 1)"
unfolding F4 F2
using A P_def assms(2) monom_closed smult_assoc1 by auto
have F6: "d ⊙⇘UP R⇙ (Cring_Poly.compose R (pderiv (up_ring.monom P a n)) (up_ring.monom (UP R) d 1)) =
(d ⊗ d[^](n-1) ⊗[n] ⋅ a) ⊙⇘UP R⇙ ((up_ring.monom (UP R) 𝟭 (n-1)))"
unfolding F5 using False A assms P_def R.m_assoc R.m_closed R.m_comm R.nat_pow_closed monic_monom_smult monom_mult_smult
by (smt (verit) R.add.nat_pow_closed)
have F7: "pderiv (Cring_Poly.compose R (up_ring.monom P a n) (up_ring.monom (UP R) d 1)) = (a ⊗ ([n]⋅⇘R⇙(d[^]n)) ⊙⇘UP R⇙ (up_ring.monom P 𝟭 (n-1)))"
unfolding F0 using A assms P_def R.m_closed R.nat_pow_closed monic_monom_smult monom_mult_smult
by simp
have F8: "a ⊗ [n] ⋅ (d [^] n) = d ⊗ d [^] (n - 1) ⊗ [n] ⋅ a"
proof-
have F80: "d ⊗ d [^] (n - 1) ⊗ [n] ⋅ a = d [^] n ⊗ [n] ⋅ a"
using A assms False by (metis R.nat_pow_Suc2 add.right_neutral add_eq_if)
show ?thesis unfolding F80
using A R.add_pow_rdistr R.m_comm R.nat_pow_closed assms(2) by presburger
qed
show ?thesis unfolding F6 F7 unfolding F8 P_def by blast
qed
qed
qed
lemma(in UP_cring) linear_sub_deriv':
assumes "f ∈ carrier (UP R)"
assumes "d ∈ carrier R"
assumes "g = compose R f (up_ring.monom (UP R) d 1)"
assumes "c ∈ carrier R"
shows "pderiv g = compose R (d ⊙⇘UP R⇙ pderiv f) (up_ring.monom (UP R) d 1)"
using assms linear_sub_deriv[of f d g c] P_def is_UP_monomE(1) is_UP_monomI pderiv_closed sub_smult by metis
lemma(in UP_cring) linear_sub_inv:
assumes "f ∈ carrier (UP R)"
assumes "d ∈ Units R"
assumes "g = compose R f (up_ring.monom (UP R) d 1)"
shows "compose R g (up_ring.monom (UP R) (inv d) 1) = f"
unfolding assms
proof fix x
have 0: "Cring_Poly.compose R (Cring_Poly.compose R f (up_ring.monom (UP R) d 1)) (up_ring.monom (UP R) (inv d) 1) x =
(inv d)[^]x ⊗ ((Cring_Poly.compose R f (up_ring.monom (UP R) d 1)) x)"
apply(rule linear_sub_cfs)
using P_def R.Units_closed assms(1) assms(2) monom_closed sub_closed apply auto[1]
apply (simp add: assms(2))
by blast
show "Cring_Poly.compose R (Cring_Poly.compose R f (up_ring.monom (UP R) d 1)) (up_ring.monom (UP R) (inv d) 1) x = f x "
unfolding 0 using linear_sub_cfs[of f d "Cring_Poly.compose R f (up_ring.monom (UP R) d 1)" x]
assms
by (smt (verit) R.Units_closed R.Units_inv_closed R.Units_l_inv R.m_assoc R.m_comm R.nat_pow_closed R.nat_pow_distrib R.nat_pow_one R.r_one UP_car_memE(1))
qed
lemma(in UP_cring) linear_sub_deg:
assumes "f ∈ carrier (UP R)"
assumes "d ∈ Units R"
assumes "g = compose R f (up_ring.monom (UP R) d 1)"
shows "deg R g = deg R f"
proof(cases "deg R f = 0")
case True
show ?thesis using assms
unfolding True assms using P_def True monom_closed
by (simp add: R.Units_closed sub_const)
next
case False
have 0: "monom (UP R) d 1 (deg R (monom (UP R) d 1)) = d"
using assms lcf_monom(2) by blast
have 1: "d[^](deg R f) ∈ Units R"
using assms(2)
by (metis Group.comm_monoid.axioms(1) R.units_comm_group R.units_of_pow comm_group_def monoid.nat_pow_closed units_of_carrier)
have 2: "f (deg R f) ≠ 𝟬"
using assms False P_def UP_cring.ltrm_rep_X_pow UP_cring_axioms deg_ltrm degree_monom by fastforce
have "deg R g = deg R f * deg R (up_ring.monom (UP R) d 1)"
unfolding assms
apply(rule cring_sub_deg[of "up_ring.monom (UP R) d 1" f] )
using assms P_def monom_closed apply blast
unfolding P_def apply(rule assms)
unfolding 0 using 2 1
by (metis R.Units_closed R.Units_l_cancel R.m_comm R.r_null R.zero_closed UP_car_memE(1) assms(1))
thus ?thesis using assms unfolding assms
by (metis (no_types, lifting) P_def R.Units_closed deg_monom deg_zero is_UP_monomE(1) linear_sub_inv monom_is_UP_monom(2) monom_zero mult.right_neutral mult_0_right sub_closed sub_const)
qed
end
section‹Lemmas About Polynomial Division›
context UP_cring
begin
subsection‹Division by Linear Terms›
definition UP_root_div where
"UP_root_div f a = (poly_shift (T⇘a⇙ f)) of (X_minus a)"
definition UP_root_rem where
"UP_root_rem f a = ctrm (T⇘a⇙ f)"
lemma UP_root_div_closed:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "UP_root_div f a ∈ carrier P"
using assms
unfolding UP_root_div_def
by (simp add: taylor_closed X_minus_closed poly_shift_closed sub_closed)
lemma rem_closed:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "UP_root_rem f a ∈ carrier P"
using assms
unfolding UP_root_rem_def
by (simp add: taylor_closed ctrm_is_poly)
lemma rem_deg:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "degree (UP_root_rem f a) = 0"
by (simp add: taylor_closed assms(1) assms(2) ctrm_degree UP_root_rem_def)
lemma remainder_theorem:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
assumes "g = UP_root_div f a"
assumes "r = UP_root_rem f a"
shows "f = r ⊕⇘P⇙ (X_minus a) ⊗⇘P⇙ g"
proof-
have "T⇘a⇙f = (ctrm (T⇘a⇙f)) ⊕⇘P⇙ X ⊗⇘P⇙ poly_shift (T⇘a⇙f)"
using poly_shift_eq[of "T⇘a⇙f"] assms taylor_closed
by blast
hence 1: "T⇘a⇙f of (X_minus a) = (ctrm (T⇘a⇙f)) ⊕⇘P⇙ (X_minus a) ⊗⇘P⇙ (poly_shift (T⇘a⇙f) of (X_minus a))"
using assms taylor_closed[of f a] X_minus_closed[of a] X_closed
sub_add[of "X_minus a" "ctrm (T⇘a⇙f)" "X ⊗⇘P⇙ poly_shift (T⇘a⇙f)"]
sub_const[of "X_minus a"] sub_mult[of "X_minus a" X "poly_shift (T⇘a⇙f)"]
ctrm_degree ctrm_is_poly P.m_closed poly_shift_closed sub_X
by presburger
have 2: "f = (ctrm (T⇘a⇙f)) ⊕⇘P⇙ (X_minus a) ⊗⇘P⇙ (poly_shift (T⇘a⇙f) of (X_minus a))"
using 1 taylor_id[of a f] assms
by simp
then show ?thesis
using assms
unfolding UP_root_div_def UP_root_rem_def
by auto
qed
lemma remainder_theorem':
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
shows "f = UP_root_rem f a ⊕⇘P⇙ (X_minus a) ⊗⇘P⇙ UP_root_div f a"
using assms remainder_theorem by auto
lemma factor_theorem:
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
assumes "g = UP_root_div f a"
assumes "to_fun f a = 𝟬"
shows "f = (X_minus a) ⊗⇘P⇙ g"
using remainder_theorem[of f a g _] assms
unfolding UP_root_rem_def
by (simp add: ctrm_zcf taylor_zcf taylor_closed UP_root_div_closed X_minus_closed)
lemma factor_theorem':
assumes "f ∈ carrier P"
assumes "a ∈ carrier R"
assumes "to_fun f a = 𝟬"
shows "f = (X_minus a) ⊗⇘P⇙ UP_root_div f a"
by (simp add: assms(1) assms(2) assms(3) factor_theorem)
subsection‹Geometric Sums›
lemma geom_quot:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
assumes "p = monom P 𝟭 (Suc n) ⊖⇘P⇙ monom P (b[^](Suc n)) 0 "
assumes "g = UP_root_div p b"
shows "a[^](Suc n) ⊖ b[^] (Suc n) = (a ⊖ b) ⊗ (to_fun g a)"
proof-
have root: "to_fun p b = 𝟬"
using assms to_fun_const[of "b[^](Suc n)" b] to_fun_monic_monom[of b "Suc n"] R.nat_pow_closed[of b "Suc n"]
to_fun_diff[of "monom P 𝟭 (Suc n)" "monom P (b[^](Suc n)) 0" b] monom_closed
by (metis P.minus_closed P_def R.one_closed R.zero_closed UP_cring.f_minus_ctrm
UP_cring.to_fun_diff UP_cring_axioms zcf_to_fun cfs_monom to_fun_const)
have LHS: "to_fun p a = a[^](Suc n) ⊖ b[^] (Suc n)"
using assms to_fun_const to_fun_monic_monom to_fun_diff
by auto
have RHS: "to_fun ((X_minus b) ⊗⇘P⇙ g) a = (a ⊖ b) ⊗ (to_fun g a)"
using to_fun_mult[of g "X_minus b"] assms X_minus_closed
by (metis P.minus_closed P_def R.nat_pow_closed R.one_closed UP_cring.UP_root_div_closed UP_cring_axioms to_fun_X_minus monom_closed)
show ?thesis
using RHS LHS root factor_theorem' assms(2) assms(3) assms(4)
by auto
qed
end
context UP_cring
begin
definition geometric_series where
"geometric_series n a b = to_fun (UP_root_div (monom P 𝟭 (Suc n) ⊖⇘UP R⇙ (monom P (b[^](Suc n)) 0)) b) a"
lemma geometric_series_id:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "a[^](Suc n) ⊖b[^] (Suc n) = (a ⊖ b) ⊗ (geometric_series n a b)"
using assms geom_quot
by (simp add: P_def geometric_series_def)
lemma geometric_series_closed:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "(geometric_series n a b) ∈ carrier R"
unfolding geometric_series_def
using assms P.minus_closed P_def UP_root_div_closed to_fun_closed monom_closed
by auto
text‹Shows that $a^n - b^n$ has $a - b$ as a factor:›
lemma to_fun_monic_monom_diff:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "∃c. c ∈ carrier R ∧ to_fun (monom P 𝟭 n) a ⊖ to_fun (monom P 𝟭 n) b = (a ⊖ b) ⊗ c"
proof(cases "n = 0")
case True
have "to_fun (monom P 𝟭 0) a ⊖ to_fun (monom P 𝟭 0) b = (a ⊖ b) ⊗ 𝟬"
unfolding a_minus_def using to_fun_const[of 𝟭] assms
by (simp add: R.r_neg)
then show ?thesis
using True by blast
next
case False
then show ?thesis
using Suc_diff_1[of n] geometric_series_id[of a b "n-1"] geometric_series_closed[of a b "n-1"]
assms(1) assms(2) to_fun_monic_monom
by auto
qed
lemma to_fun_diff_factor:
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
assumes "f ∈ carrier P"
shows "∃c. c ∈ carrier R ∧(to_fun f a) ⊖ (to_fun f b) = (a ⊖ b)⊗c"
proof(rule poly_induct5[of f])
show "f ∈ carrier P" using assms by simp
show "⋀p q. q ∈ carrier P ⟹
p ∈ carrier P ⟹
∃c. c ∈ carrier R ∧ to_fun p a ⊖ to_fun p b = (a ⊖ b) ⊗ c ⟹
∃c. c ∈ carrier R ∧ to_fun q a ⊖ to_fun q b = (a ⊖ b) ⊗ c ⟹
∃c. c ∈ carrier R ∧ to_fun (p ⊕⇘P⇙ q) a ⊖ to_fun (p ⊕⇘P⇙ q) b = (a ⊖ b) ⊗ c"
proof- fix p q assume A: "q ∈ carrier P" "p ∈ carrier P"
"∃c. c ∈ carrier R ∧ to_fun p a ⊖ to_fun p b = (a ⊖ b) ⊗ c"
"∃c. c ∈ carrier R ∧ to_fun q a ⊖ to_fun q b = (a ⊖ b) ⊗ c"
obtain c where c_def: "c ∈ carrier R ∧ to_fun p a ⊖ to_fun p b = (a ⊖ b) ⊗ c"
using A by blast
obtain c' where c'_def: "c' ∈ carrier R ∧ to_fun q a ⊖ to_fun q b = (a ⊖ b) ⊗ c'"
using A by blast
have 0: "(a ⊖ b) ⊗ c ⊕ (a ⊖ b) ⊗ c' = (a ⊖ b)⊗(c ⊕ c')"
using assms c_def c'_def unfolding a_minus_def
by (simp add: R.r_distr R.r_minus)
have 1: "to_fun (p ⊕⇘P⇙q) a ⊖ to_fun (p ⊕⇘P⇙ q) b = to_fun p a ⊕ to_fun q a ⊖ to_fun p b ⊖ to_fun q b"
using A to_fun_plus[of p q a] to_fun_plus[of p q b] assms to_fun_closed
R.ring_simprules(19)[of "to_fun p b" "to_fun q b"]
by (simp add: R.add.m_assoc R.minus_eq to_fun_plus)
hence "to_fun (p ⊕⇘P⇙q) a ⊖ to_fun (p ⊕⇘P⇙ q) b = to_fun p a ⊖ to_fun p b ⊕ to_fun q a ⊖ to_fun q b"
using 0 A assms R.ring_simprules to_fun_closed a_assoc a_comm
unfolding a_minus_def by (smt (verit, del_insts))
hence "to_fun (p ⊕⇘P⇙q) a ⊖ to_fun (p ⊕⇘P⇙ q) b = to_fun p a ⊖ to_fun p b ⊕ (to_fun q a ⊖ to_fun q b)"
using 0 A assms R.ring_simprules to_fun_closed
unfolding a_minus_def by metis
hence "to_fun (p ⊕⇘P⇙q) a ⊖ to_fun (p ⊕⇘P⇙ q) b = (a ⊖ b)⊗(c ⊕ c')"
using 0 A c_def c'_def
by simp
thus "∃c. c ∈ carrier R ∧ to_fun (p ⊕⇘P⇙ q) a ⊖ to_fun (p ⊕⇘P⇙ q) b = (a ⊖ b) ⊗ c"
using R.add.m_closed c'_def c_def by blast
qed
show "⋀n. ∃c. c ∈ carrier R ∧ to_fun (monom P 𝟭 n) a ⊖ to_fun (monom P 𝟭 n) b = (a ⊖ b) ⊗ c"
by (simp add: assms(1) assms(2) to_fun_monic_monom_diff)
show "⋀p aa.
aa ∈ carrier R ⟹
p ∈ carrier P ⟹ ∃c. c ∈ carrier R ∧ to_fun p a ⊖ to_fun p b = (a ⊖ b) ⊗ c ⟹ ∃c. c ∈ carrier R ∧ to_fun (aa ⊙⇘P⇙ p) a ⊖ to_fun (aa ⊙⇘P⇙ p) b = (a ⊖ b) ⊗ c"
proof- fix p c assume A: "c ∈ carrier R" " p ∈ carrier P"
"∃e. e ∈ carrier R ∧ to_fun p a ⊖ to_fun p b = (a ⊖ b) ⊗ e"
then obtain d where d_def: "d ∈ carrier R ∧ to_fun p a ⊖ to_fun p b = (a ⊖ b) ⊗ d"
by blast
have "to_fun (c ⊙⇘P⇙ p) a ⊖ to_fun (c ⊙⇘P⇙ p) b = c ⊗ (to_fun p a ⊖ to_fun p b)"
using A d_def assms to_fun_smult[of p a c] to_fun_smult[of p b c]
to_fun_closed[of p a] to_fun_closed[of p b] R.ring_simprules
by presburger
hence "c⊗d ∈ carrier R ∧ to_fun (c ⊙⇘P⇙ p) a ⊖ to_fun (c ⊙⇘P⇙ p) b = (a ⊖ b) ⊗ (c ⊗d)"
by (simp add: A(1) R.m_lcomm assms(1) assms(2) d_def)
thus "∃e. e ∈ carrier R ∧ to_fun (c ⊙⇘P⇙ p) a ⊖ to_fun (c ⊙⇘P⇙ p) b = (a ⊖ b) ⊗ e"
by blast
qed
qed
text‹Any finite set over a domain is the zero set of a polynomial:›
lemma(in UP_domain) fin_set_poly_roots:
assumes "F ⊆ carrier R"
assumes "finite F"
shows "∃ P ∈ carrier (UP R). ∀ x ∈ carrier R. to_fun P x = 𝟬 ⟷ x ∈ F"
apply(rule finite.induct)
apply (simp add: assms(2))
proof-
show "∃P∈carrier (UP R). ∀x∈carrier R. (to_fun P x = 𝟬) = (x ∈ {})"
proof-
have "∀x∈carrier R. (to_fun (𝟭⇘UP R⇙) x = 𝟬) = (x ∈ {})"
proof
fix x
assume A: "x ∈ carrier R"
then have "(to_fun (𝟭⇘UP R⇙)) x = 𝟭"
by (metis P_def R.one_closed UP_cring.to_fun_to_poly UP_cring_axioms ring_hom_one to_poly_is_ring_hom)
then show "(to_fun 𝟭⇘UP R⇙ x = 𝟬) = (x ∈ {})"
by simp
qed
then show ?thesis
using P_def UP_one_closed
by blast
qed
show "⋀A a. finite A ⟹
∃P∈carrier (UP R). ∀x∈carrier R. (to_fun P x = 𝟬) = (x ∈ A) ⟹ ∃P∈carrier (UP R). ∀x∈carrier R. (to_fun P x = 𝟬) = (x ∈ insert a A)"
proof-
fix A :: "'a set" fix a
assume fin_A: "finite A"
assume IH: "∃P∈carrier (UP R). ∀x∈carrier R. (to_fun P x = 𝟬) = (x ∈ A)"
then obtain p where p_def: "p ∈carrier (UP R) ∧ (∀x∈carrier R. (to_fun p x = 𝟬) = (x ∈ A))"
by blast
show "∃P∈carrier (UP R). ∀x∈carrier R. (to_fun P x = 𝟬) = (x ∈ insert a A)"
proof(cases "a ∈ carrier R")
case True
obtain Q where Q_def: "Q = p ⊗⇘UP R⇙ (X ⊖⇘UP R⇙ to_poly a)"
by blast
have "∀x∈carrier R. (to_fun Q x = 𝟬) = (x ∈ insert a A)"
proof fix x
assume P: "x ∈ carrier R"
have P0: "to_fun (X ⊖⇘UP R⇙ to_poly a) x = x ⊖ a"
using to_fun_plus[of X "⊖⇘UP R⇙ to_poly a" x] True P
unfolding a_minus_def
by (metis X_poly_minus_def a_minus_def to_fun_X_minus)
then have "to_fun Q x = (to_fun p x) ⊗ (x ⊖ a)"
proof-
have 0: " p ∈ carrier P"
by (simp add: P_def p_def)
have 1: " X ⊖⇘UP R⇙ to_poly a ∈ carrier P"
using P.minus_closed P_def True X_closed to_poly_closed by auto
have 2: "x ∈ carrier R"
by (simp add: P)
then show ?thesis
using to_fun_mult[of p "(X ⊖⇘UP R⇙ to_poly a)" x] P0 0 1 2 Q_def True P_def to_fun_mult
by auto
qed
then show "(to_fun Q x = 𝟬) = (x ∈ insert a A)"
using p_def
by (metis P R.add.inv_closed R.integral_iff R.l_neg R.minus_closed R.minus_unique True UP_cring.to_fun_closed UP_cring_axioms a_minus_def insert_iff)
qed
then have "Q ∈ carrier (UP R) ∧ (∀x∈carrier R. (to_fun Q x = 𝟬) = (x ∈ insert a A))"
using P.minus_closed P_def Q_def True UP_mult_closed X_closed p_def to_poly_closed by auto
then show ?thesis
by blast
next
case False
then show ?thesis
using IH subsetD by auto
qed
qed
qed
subsection‹Polynomial Evaluation at Multiplicative Inverses›
text‹For every polynomial $p(x)$ of degree $n$, there is a unique polynomial $q(x)$ which satisfies the equation $q(x) = x^n p(1/x)$. This section defines this polynomial and proves this identity.›
definition(in UP_cring) one_over_poly where
"one_over_poly p = (λ n. if n ≤ degree p then p ((degree p) - n) else 𝟬)"
lemma(in UP_cring) one_over_poly_closed:
assumes "p ∈ carrier P"
shows "one_over_poly p ∈ carrier P"
apply(rule UP_car_memI[of "degree p" ])
unfolding one_over_poly_def using assms apply simp
by (simp add: assms cfs_closed)
lemma(in UP_cring) one_over_poly_monom:
assumes "a ∈ carrier R"
shows "one_over_poly (monom P a n) = monom P a 0"
apply(rule ext)
unfolding one_over_poly_def using assms
by (metis cfs_monom deg_monom diff_diff_cancel diff_is_0_eq diff_self_eq_0 zero_diff)
lemma(in UP_cring) one_over_poly_monom_add:
assumes "a ∈ carrier R"
assumes "a ≠ 𝟬"
assumes "p ∈ carrier P"
assumes "degree p < n"
shows "one_over_poly (p ⊕⇘P⇙ monom P a n) = monom P a 0 ⊕⇘P⇙ monom P 𝟭 (n - degree p) ⊗⇘P⇙ one_over_poly p"
proof-
have 0: "degree (p ⊕⇘P⇙ monom P a n) = n"
by (simp add: assms(1) assms(2) assms(3) assms(4) equal_deg_sum)
show ?thesis
proof(rule ext) fix x show "one_over_poly (p ⊕⇘P⇙ monom P a n) x =
(monom P a 0 ⊕⇘P⇙ monom P 𝟭 (n - deg R p) ⊗⇘P⇙ one_over_poly p) x"
proof(cases "x = 0")
case T: True
have T0: "one_over_poly (p ⊕⇘P⇙ monom P a n) 0 = a"
unfolding one_over_poly_def
by (metis lcf_eq lcf_monom(1) ltrm_of_sum_diff_deg P.add.m_closed assms(1) assms(2) assms(3) assms(4) diff_zero le0 monom_closed)
have T1: "(monom P a 0 ⊕⇘P⇙ monom P 𝟭 (n - degree p) ⊗⇘P⇙ one_over_poly p) 0 = a"
using one_over_poly_closed
by (metis (no_types, lifting) lcf_monom(1) R.one_closed R.r_zero UP_m_comm UP_mult_closed assms(1) assms(3) assms(4) cfs_add cfs_monom_mult deg_const monom_closed zero_less_diff)
show ?thesis using T0 T1 T by auto
next
case F: False
show ?thesis
proof(cases "x < n - degree p")
case True
then have T0: "degree p < n - x ∧ n - x < n"
using F by auto
then have T1: "one_over_poly (p ⊕⇘P⇙ monom P a n) x = 𝟬"
using True F 0 unfolding one_over_poly_def
using assms(1) assms(3) coeff_of_sum_diff_degree0
by (metis ltrm_cfs ltrm_of_sum_diff_deg P.add.m_closed P.add.m_comm assms(2) assms(4) monom_closed nat_neq_iff)
have "(monom P a 0 ⊕⇘P⇙ monom P 𝟭 (n - degree p) ⊗⇘P⇙ one_over_poly p) x = 𝟬"
using True F 0 one_over_poly_def one_over_poly_closed
by (metis (no_types, lifting) P.add.m_comm P.m_closed R.one_closed UP_m_comm assms(1)
assms(3) cfs_monom_mult coeff_of_sum_diff_degree0 deg_const monom_closed neq0_conv)
then show ?thesis using T1 by auto
next
case False
then have "n - degree p ≤ x"
by auto
then obtain k where k_def: "k + (n - degree p) = x"
using le_Suc_ex diff_add
by blast
have F0: "(monom P a 0 ⊕⇘P⇙ monom P 𝟭 (n - deg R p) ⊗⇘P⇙ one_over_poly p) x
= one_over_poly p k"
using k_def one_over_poly_closed assms
times_X_pow_coeff[of "one_over_poly p" "n - deg R p" k]
P.m_closed
by (metis (no_types, lifting) P.add.m_comm R.one_closed add_gr_0 coeff_of_sum_diff_degree0 deg_const monom_closed zero_less_diff)
show ?thesis
proof(cases "x ≤ n")
case True
have T0: "n - x = degree p - k"
using assms(4) k_def by linarith
have T1: "n - x < n"
using True F
by linarith
then have F1: "(p ⊕⇘P⇙ monom P a n) (n - x) = p (degree p - k)"
using True False F0 0 k_def cfs_add
by (simp add: F0 T0 assms(1) assms(3) cfs_closed cfs_monom)
then show ?thesis
using "0" F0 assms(1) assms(2) assms(3) degree_of_sum_diff_degree k_def one_over_poly_def
by auto
next
case False
then show ?thesis
using "0" F0 assms(1) assms(2) assms(3) degree_of_sum_diff_degree k_def one_over_poly_def
by auto
qed
qed
qed
qed
qed
lemma( in UP_cring) one_over_poly_eval:
assumes "p ∈ carrier P"
assumes "x ∈ carrier R"
assumes "x ∈ Units R"
shows "to_fun (one_over_poly p) x = (x[^](degree p)) ⊗ (to_fun p (inv⇘R⇙ x))"
proof(rule poly_induct6[of p])
show " p ∈ carrier P"
using assms by simp
show "⋀a n. a ∈ carrier R ⟹
to_fun (one_over_poly (monom P a 0)) x = x [^] deg R (monom P a 0) ⊗ to_fun (monom P a 0) (inv x)"
using assms to_fun_const one_over_poly_monom by auto
show "⋀a n p.
a ∈ carrier R ⟹
a ≠ 𝟬 ⟹
p ∈ carrier P ⟹
deg R p < n ⟹
to_fun (one_over_poly p) x = x [^] deg R p ⊗ to_fun p (inv x) ⟹
to_fun (one_over_poly (p ⊕⇘P⇙ monom P a n)) x = x [^] deg R (p ⊕⇘P⇙ monom P a n) ⊗ to_fun (p ⊕⇘P⇙ monom P a n) (inv x)"
proof- fix a n p assume A: "a ∈ carrier R" "a ≠ 𝟬" "p ∈ carrier P" "deg R p < n"
"to_fun (one_over_poly p) x = x [^] deg R p ⊗ to_fun p (inv x)"
have "one_over_poly (p ⊕⇘P⇙ monom P a n) = monom P a 0 ⊕⇘P⇙ monom P 𝟭 (n - degree p) ⊗⇘P⇙ one_over_poly p"
using A by (simp add: one_over_poly_monom_add)
hence "to_fun ( one_over_poly (p ⊕⇘P⇙ monom P a n)) x =
a ⊕ to_fun ( monom P 𝟭 (n - degree p) ⊗⇘P⇙ one_over_poly p) x"
using A to_fun_plus one_over_poly_closed cfs_add
by (simp add: assms(2) to_fun_const)
hence "to_fun (one_over_poly (p ⊕⇘P⇙ monom P a n)) x = a ⊕ x[^](n - degree p) ⊗ x [^] degree p ⊗ to_fun p (inv x)"
by (simp add: A(3) A(5) R.m_assoc assms(2) assms(3) to_fun_closed to_fun_monic_monom to_fun_mult one_over_poly_closed)
hence 0:"to_fun (one_over_poly (p ⊕⇘P⇙ monom P a n)) x = a ⊕ x[^]n ⊗ to_fun p (inv x)"
using A R.nat_pow_mult assms(2)
by auto
have 1: "to_fun (one_over_poly (p ⊕⇘P⇙ monom P a n)) x = x[^]n ⊗ (a ⊗ inv x [^]n ⊕ to_fun p (inv x))"
proof-
have "x[^]n ⊗ a ⊗ inv x [^]n = a"
by (metis (no_types, opaque_lifting) A(1) R.Units_inv_closed R.Units_r_inv R.m_assoc
R.m_comm R.nat_pow_closed R.nat_pow_distrib R.nat_pow_one R.r_one assms(2) assms(3))
thus ?thesis
using A R.ring_simprules(23)[of _ _ "x[^]n"] 0 R.m_assoc assms(2) assms(3) to_fun_closed
by auto
qed
have 2: "degree (p ⊕⇘P⇙ monom P a n) = n"
by (simp add: A(1) A(2) A(3) A(4) equal_deg_sum)
show " to_fun (one_over_poly (p ⊕⇘P⇙ monom P a n)) x = x [^] deg R (p ⊕⇘P⇙ monom P a n) ⊗ to_fun (p ⊕⇘P⇙ monom P a n) (inv x)"
using 1 2
by (metis (no_types, lifting) A(1) A(3) P_def R.Units_inv_closed R.add.m_comm
UP_cring.to_fun_monom UP_cring_axioms assms(3) to_fun_closed to_fun_plus monom_closed)
qed
qed
end
section‹Lifting Homomorphisms of Rings to Polynomial Rings by Application to Coefficients›
definition poly_lift_hom where
"poly_lift_hom R S φ = eval R (UP S) (to_polynomial S ∘ φ) (X_poly S)"
context UP_ring
begin
lemma(in UP_cring) pre_poly_lift_hom_is_hom:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
shows "ring_hom_ring R (UP S) (to_polynomial S ∘ φ)"
apply(rule ring_hom_ringI)
apply (simp add: R.ring_axioms)
apply (simp add: UP_ring.UP_ring UP_ring.intro assms(1) cring.axioms(1))
using UP_cring.intro UP_cring.to_poly_closed assms(1) assms(2) ring_hom_closed apply fastforce
using assms UP_cring.to_poly_closed[of S] ring_hom_closed[of φ R S] comp_apply[of "to_polynomial S" φ]
unfolding UP_cring_def
apply (metis UP_cring.to_poly_mult UP_cring_def ring_hom_mult)
using assms UP_cring.to_poly_closed[of S] ring_hom_closed[of φ R S] comp_apply[of "to_polynomial S" φ]
unfolding UP_cring_def
apply (metis UP_cring.to_poly_add UP_cring_def ring_hom_add)
using assms UP_cring.to_poly_closed[of S] ring_hom_one[of φ R S] comp_apply[of "to_polynomial S" φ]
unfolding UP_cring_def
by (simp add: ‹φ ∈ ring_hom R S ⟹ φ 𝟭 = 𝟭⇘S⇙› UP_cring.intro UP_cring.to_poly_is_ring_hom ring_hom_one)
lemma(in UP_cring) poly_lift_hom_is_hom:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
shows "poly_lift_hom R S φ ∈ ring_hom (UP R) (UP S)"
unfolding poly_lift_hom_def
apply( rule UP_pre_univ_prop.eval_ring_hom[of R "UP S" ])
unfolding UP_pre_univ_prop_def
apply (simp add: R_cring RingHom.ring_hom_cringI UP_cring.UP_cring UP_cring_def assms(1) assms(2) pre_poly_lift_hom_is_hom)
by (simp add: UP_cring.X_closed UP_cring.intro assms(1))
lemma(in UP_cring) poly_lift_hom_closed:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "p ∈ carrier (UP R)"
shows "poly_lift_hom R S φ p ∈ carrier (UP S)"
by (metis assms(1) assms(2) assms(3) poly_lift_hom_is_hom ring_hom_closed)
lemma(in UP_cring) poly_lift_hom_add:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "p ∈ carrier (UP R)"
assumes "q ∈ carrier (UP R)"
shows "poly_lift_hom R S φ (p ⊕⇘UP R⇙ q) = poly_lift_hom R S φ p ⊕⇘UP S⇙ poly_lift_hom R S φ q"
using assms poly_lift_hom_is_hom[of S φ] ring_hom_add[of "poly_lift_hom R S φ" "UP R" "UP S" p q]
by blast
lemma(in UP_cring) poly_lift_hom_mult:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "p ∈ carrier (UP R)"
assumes "q ∈ carrier (UP R)"
shows "poly_lift_hom R S φ (p ⊗⇘UP R⇙ q) = poly_lift_hom R S φ p ⊗⇘UP S⇙ poly_lift_hom R S φ q"
using assms poly_lift_hom_is_hom[of S φ] ring_hom_mult[of "poly_lift_hom R S φ" "UP R" "UP S" p q]
by blast
lemma(in UP_cring) poly_lift_hom_extends_hom:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "r ∈ carrier R"
shows "poly_lift_hom R S φ (to_polynomial R r) = to_polynomial S (φ r)"
using UP_pre_univ_prop.eval_const[of R "UP S" "to_polynomial S ∘ φ" "X_poly S" r ] assms
comp_apply[of "λa. monom (UP S) a 0" φ r] pre_poly_lift_hom_is_hom[of S φ]
unfolding poly_lift_hom_def to_polynomial_def UP_pre_univ_prop_def
by (simp add: R_cring RingHom.ring_hom_cringI UP_cring.UP_cring UP_cring.X_closed UP_cring.intro)
lemma(in UP_cring) poly_lift_hom_extends_hom':
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "r ∈ carrier R"
shows "poly_lift_hom R S φ (monom P r 0) = monom (UP S) (φ r) 0"
using poly_lift_hom_extends_hom[of S φ r] assms
unfolding to_polynomial_def P_def
by blast
lemma(in UP_cring) poly_lift_hom_smult:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "p ∈ carrier (UP R)"
assumes "a ∈ carrier R"
shows "poly_lift_hom R S φ (a ⊙⇘UP R⇙ p) = φ a ⊙⇘UP S⇙ (poly_lift_hom R S φ p)"
using assms poly_lift_hom_is_hom[of S φ] poly_lift_hom_extends_hom'[of S φ a]
poly_lift_hom_mult[of S φ "monom P a 0" p] ring_hom_closed[of φ R S a]
UP_ring.monom_mult_is_smult[of S "φ a" "poly_lift_hom R S φ p"]
monom_mult_is_smult[of a p] monom_closed[of a 0] poly_lift_hom_closed[of S φ p]
unfolding to_polynomial_def UP_ring_def P_def cring_def
by simp
lemma(in UP_cring) poly_lift_hom_monom:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "r ∈ carrier R"
shows "poly_lift_hom R S φ (monom (UP R) r n) = (monom (UP S) (φ r) n)"
proof-
have "eval R (UP S) (to_polynomial S ∘ φ) (X_poly S) (monom (UP R) r n) = (to_polynomial S ∘ φ) r ⊗⇘UP S⇙ X_poly S [^]⇘UP S⇙ n"
using assms UP_pre_univ_prop.eval_monom[of R "UP S" "to_polynomial S ∘ φ" r "X_poly S" n]
unfolding UP_pre_univ_prop_def UP_cring_def ring_hom_cring_def
by (meson UP_cring.UP_cring UP_cring.X_closed UP_cring.pre_poly_lift_hom_is_hom UP_cring_axioms
UP_cring_def ring_hom_cring_axioms.intro ring_hom_ring.homh)
then have "eval R (UP S) (to_polynomial S ∘ φ) (X_poly S) (monom (UP R) r n) = (to_polynomial S (φ r)) ⊗⇘UP S⇙ X_poly S [^]⇘UP S⇙ n"
by simp
then show ?thesis
unfolding poly_lift_hom_def
using assms UP_cring.monom_rep_X_pow[of S "φ r" n] ring_hom_closed[of φ R S r]
by (metis UP_cring.X_closed UP_cring.intro UP_cring.monom_sub UP_cring.sub_monom(1))
qed
lemma(in UP_cring) poly_lift_hom_X_var:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
shows "poly_lift_hom R S φ (monom (UP R) 𝟭⇘R⇙ 1) = (monom (UP S) 𝟭⇘S⇙ 1)"
using assms(1) assms(2) poly_lift_hom_monom ring_hom_one by fastforce
lemma(in UP_cring) poly_lift_hom_X_var':
assumes "cring S"
assumes "φ ∈ ring_hom R S"
shows "poly_lift_hom R S φ (X_poly R) = (X_poly S)"
unfolding X_poly_def
using assms(1) assms(2) poly_lift_hom_X_var by blast
lemma(in UP_cring) poly_lift_hom_X_var'':
assumes "cring S"
assumes "φ ∈ ring_hom R S"
shows "poly_lift_hom R S φ (monom (UP R) 𝟭⇘R⇙ n) = (monom (UP S) 𝟭⇘S⇙ n)"
using assms(1) assms(2) poly_lift_hom_monom ring_hom_one by fastforce
lemma(in UP_cring) poly_lift_hom_X_var''':
assumes "cring S"
assumes "φ ∈ ring_hom R S"
shows "poly_lift_hom R S φ (X_poly R [^]⇘UP R⇙ (n::nat)) = (X_poly S) [^]⇘UP S⇙ (n::nat)"
using assms
by (smt (verit) ltrm_of_X P.nat_pow_closed P_def R.ring_axioms UP_cring.to_fun_closed UP_cring.intro
UP_cring.monom_pow UP_cring.poly_lift_hom_monom UP_cring_axioms X_closed cfs_closed
cring.axioms(1) to_fun_X_pow poly_lift_hom_X_var' ring_hom_closed ring_hom_nat_pow)
lemma(in UP_cring) poly_lift_hom_X_plus:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "a ∈ carrier R"
shows "poly_lift_hom R S φ (X_poly_plus R a) = X_poly_plus S (φ a)"
using ring_hom_add
unfolding X_poly_plus_def
using P_def X_closed assms(1) assms(2) assms(3) poly_lift_hom_X_var' poly_lift_hom_add poly_lift_hom_extends_hom to_poly_closed by fastforce
lemma(in UP_cring) poly_lift_hom_X_plus_nat_pow:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "a ∈ carrier R"
shows "poly_lift_hom R S φ (X_poly_plus R a [^]⇘UP R⇙ (n::nat)) = X_poly_plus S (φ a) [^]⇘UP S⇙ (n::nat)"
using assms poly_lift_hom_X_plus[of S φ a]
ring_hom_nat_pow[of "UP R" "UP S" "poly_lift_hom R S φ" "X_poly_plus R a" n]
poly_lift_hom_is_hom[of S φ] X_plus_closed[of a] UP_ring.UP_ring[of S]
unfolding P_def cring_def UP_cring_def
using P_def UP_ring UP_ring.intro
by (simp add: UP_ring.intro)
lemma(in UP_cring) X_poly_plus_nat_pow_closed:
assumes "a ∈ carrier R"
shows " X_poly_plus R a [^]⇘UP R⇙ (n::nat) ∈ carrier (UP R)"
using assms P.nat_pow_closed P_def X_plus_closed by auto
lemma(in UP_cring) poly_lift_hom_X_plus_nat_pow_smult:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "poly_lift_hom R S φ (b ⊙⇘UP R⇙ X_poly_plus R a [^]⇘UP R⇙ (n::nat)) = φ b ⊙⇘UP S ⇙X_poly_plus S (φ a) [^]⇘UP S⇙ (n::nat)"
by (simp add: X_poly_plus_nat_pow_closed assms(1) assms(2) assms(3) assms(4) poly_lift_hom_X_plus_nat_pow poly_lift_hom_smult)
lemma(in UP_cring) poly_lift_hom_X_minus:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "a ∈ carrier R"
shows "poly_lift_hom R S φ (X_poly_minus R a) = X_poly_minus S (φ a)"
using poly_lift_hom_X_plus[of S φ "⊖ a"] X_minus_plus[of a] UP_cring.X_minus_plus[of S "φ a"]
R.ring_hom_a_inv[of S φ a]
unfolding UP_cring_def P_def
by (metis R.add.inv_closed assms(1) assms(2) assms(3) cring.axioms(1) ring_hom_closed)
lemma(in UP_cring) poly_lift_hom_X_minus_nat_pow:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "a ∈ carrier R"
shows "poly_lift_hom R S φ (X_poly_minus R a [^]⇘UP R⇙ (n::nat)) = X_poly_minus S (φ a) [^]⇘UP S⇙ (n::nat)"
using assms poly_lift_hom_X_minus ring_hom_nat_pow X_minus_plus UP_cring.X_minus_plus
poly_lift_hom_X_plus poly_lift_hom_X_plus_nat_pow by fastforce
lemma(in UP_cring) X_poly_minus_nat_pow_closed:
assumes "a ∈ carrier R"
shows "X_poly_minus R a [^]⇘UP R⇙ (n::nat) ∈ carrier (UP R)"
using assms monoid.nat_pow_closed[of "UP R" "X_poly_minus R a" n]
P.nat_pow_closed P_def X_minus_closed by auto
lemma(in UP_cring) poly_lift_hom_X_minus_nat_pow_smult:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "a ∈ carrier R"
assumes "b ∈ carrier R"
shows "poly_lift_hom R S φ (b ⊙⇘UP R⇙ X_poly_minus R a [^]⇘UP R⇙ (n::nat)) = φ b ⊙⇘UP S ⇙X_poly_minus S (φ a) [^]⇘UP S⇙ (n::nat)"
by (simp add: X_poly_minus_nat_pow_closed assms(1) assms(2) assms(3) assms(4) poly_lift_hom_X_minus_nat_pow poly_lift_hom_smult)
lemma(in UP_cring) poly_lift_hom_cf:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "p ∈ carrier P"
shows "poly_lift_hom R S φ p k = φ (p k)"
apply(rule poly_induct3[of p])
apply (simp add: assms(3))
proof-
show "⋀p q. q ∈ carrier P ⟹
p ∈ carrier P ⟹
poly_lift_hom R S φ p k = φ (p k) ⟹ poly_lift_hom R S φ q k = φ (q k) ⟹ poly_lift_hom R S φ (p ⊕⇘P⇙ q) k = φ ((p ⊕⇘P⇙ q) k)"
proof- fix p q assume A: "p ∈ carrier P" "q ∈ carrier P"
"poly_lift_hom R S φ p k = φ (p k)" "poly_lift_hom R S φ q k = φ (q k)"
show "poly_lift_hom R S φ q k = φ (q k) ⟹ poly_lift_hom R S φ (p ⊕⇘P⇙ q) k = φ ((p ⊕⇘P⇙ q) k)"
using A assms poly_lift_hom_add[of S φ p q]
poly_lift_hom_closed[of S φ p] poly_lift_hom_closed[of S φ q]
UP_ring.cfs_closed[of S "poly_lift_hom R S φ q " k] UP_ring.cfs_closed[of S "poly_lift_hom R S φ p" k]
UP_ring.cfs_add[of S "poly_lift_hom R S φ p" "poly_lift_hom R S φ q" k]
unfolding P_def UP_ring_def
by (metis (full_types) P_def cfs_add cfs_closed cring.axioms(1) ring_hom_add)
qed
show "⋀a n. a ∈ carrier R ⟹ poly_lift_hom R S φ (monom P a n) k = φ (monom P a n k)"
proof- fix a m assume A: "a ∈ carrier R"
show "poly_lift_hom R S φ (monom P a m) k = φ (monom P a m k)"
apply(cases "m = k")
using cfs_monom[of a m k] assms poly_lift_hom_monom[of S φ a m] UP_ring.cfs_monom[of S "φ a" m k]
unfolding P_def UP_ring_def
apply (simp add: A cring.axioms(1) ring_hom_closed)
using cfs_monom[of a m k] assms poly_lift_hom_monom[of S φ a m] UP_ring.cfs_monom[of S "φ a" m k]
unfolding P_def UP_ring_def
by (metis A P_def R.ring_axioms cring.axioms(1) ring_hom_closed ring_hom_zero)
qed
qed
lemma(in ring) ring_hom_monom_term:
assumes "a ∈ carrier R"
assumes "c ∈ carrier R"
assumes "ring S"
assumes "h ∈ ring_hom R S"
shows "h (a ⊗ c[^](n::nat)) = h a ⊗⇘S⇙ (h c)[^]⇘S⇙n"
apply(induction n)
using assms ringE(2) ring_hom_closed apply fastforce
by (metis assms(1) assms(2) assms(3) assms(4) local.ring_axioms nat_pow_closed ring_hom_mult ring_hom_nat_pow)
lemma(in UP_cring) poly_lift_hom_eval:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
shows "UP_cring.to_fun S (poly_lift_hom R S φ p) (φ a) = φ (to_fun p a) "
apply(rule poly_induct3[of p])
apply (simp add: assms(3))
proof-
show "⋀p q. q ∈ carrier P ⟹
p ∈ carrier P ⟹
UP_cring.to_fun S (poly_lift_hom R S φ p) (φ a) = φ (to_fun p a) ⟹
UP_cring.to_fun S (poly_lift_hom R S φ q) (φ a) = φ (to_fun q a) ⟹
UP_cring.to_fun S (poly_lift_hom R S φ (p ⊕⇘P⇙ q)) (φ a) = φ (to_fun (p ⊕⇘P⇙ q) a)"
proof- fix p q assume A: "q ∈ carrier P" "p ∈ carrier P"
"UP_cring.to_fun S (poly_lift_hom R S φ p) (φ a) = φ (to_fun p a)"
"UP_cring.to_fun S (poly_lift_hom R S φ q) (φ a) = φ (to_fun q a)"
have "(poly_lift_hom R S φ (p ⊕⇘P⇙ q)) = poly_lift_hom R S φ p ⊕⇘UP S⇙ poly_lift_hom R S φ q"
using A(1) A(2) P_def assms(1) assms(2) poly_lift_hom_add by auto
hence "UP_cring.to_fun S (poly_lift_hom R S φ (p ⊕⇘P⇙ q)) (φ a) =
UP_cring.to_fun S (poly_lift_hom R S φ p) (φ a) ⊕⇘S⇙ UP_cring.to_fun S (poly_lift_hom R S φ q) (φ a)"
using UP_cring.to_fun_plus[of S] assms
unfolding UP_cring_def
by (metis (no_types, lifting) A(1) A(2) P_def poly_lift_hom_closed ring_hom_closed)
thus "UP_cring.to_fun S (poly_lift_hom R S φ (p ⊕⇘P⇙ q)) (φ a) = φ (to_fun (p ⊕⇘P⇙ q) a)"
using A to_fun_plus assms ring_hom_add[of φ R S]
poly_lift_hom_closed[of S φ] UP_cring.to_fun_def[of S] to_fun_def
unfolding P_def UP_cring_def
using UP_cring.to_fun_closed UP_cring_axioms
by metis
qed
show "⋀c n. c ∈ carrier R ⟹ UP_cring.to_fun S (poly_lift_hom R S φ (monom P c n)) (φ a) = φ (to_fun (monom P c n) a)"
unfolding P_def
proof - fix c n assume A: "c ∈ carrier R"
have 0: "φ (a [^]⇘R⇙ (n::nat)) = φ a [^]⇘S⇙ n"
using assms ring_hom_nat_pow[of R S φ a n]
unfolding cring_def
using R.ring_axioms by blast
have 1: "φ (c ⊗⇘R⇙ a [^]⇘R⇙ n) = φ c ⊗⇘S⇙ φ a [^]⇘S⇙ n"
using ring_hom_mult[of φ R S c "a [^]⇘R⇙ n" ] 0 assms A monoid.nat_pow_closed [of R a n]
by (simp add: cring.axioms(1) ringE(2))
show "UP_cring.to_fun S (poly_lift_hom R S φ (monom (UP R) c n)) (φ a) = φ (to_fun(monom (UP R) c n) a)"
using assms A poly_lift_hom_monom[of S φ c n] UP_cring.to_fun_monom[of S "φ c" "φ a" n]
to_fun_monom[of c a n] 0 1 ring_hom_closed[of φ R S] unfolding UP_cring_def
by (simp add: P_def to_fun_def)
qed
qed
lemma(in UP_cring) poly_lift_hom_sub:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "p ∈ carrier P"
assumes "q ∈ carrier P"
shows "poly_lift_hom R S φ (compose R p q) = compose S (poly_lift_hom R S φ p) (poly_lift_hom R S φ q)"
apply(rule poly_induct3[of p])
apply (simp add: assms(3))
proof-
show " ⋀p qa.
qa ∈ carrier P ⟹
p ∈ carrier P ⟹
poly_lift_hom R S φ (Cring_Poly.compose R p q) = Cring_Poly.compose S (poly_lift_hom R S φ p) (poly_lift_hom R S φ q) ⟹
poly_lift_hom R S φ (Cring_Poly.compose R qa q) = Cring_Poly.compose S (poly_lift_hom R S φ qa) (poly_lift_hom R S φ q) ⟹
poly_lift_hom R S φ (Cring_Poly.compose R (p ⊕⇘P⇙ qa) q) = Cring_Poly.compose S (poly_lift_hom R S φ (p ⊕⇘P⇙ qa)) (poly_lift_hom R S φ q)"
proof- fix a b assume A: "a ∈ carrier P"
"b ∈ carrier P"
"poly_lift_hom R S φ (Cring_Poly.compose R a q) = Cring_Poly.compose S (poly_lift_hom R S φ a) (poly_lift_hom R S φ q)"
"poly_lift_hom R S φ (Cring_Poly.compose R b q) = Cring_Poly.compose S (poly_lift_hom R S φ b) (poly_lift_hom R S φ q)"
show "poly_lift_hom R S φ (Cring_Poly.compose R (a ⊕⇘P⇙ b) q) = Cring_Poly.compose S (poly_lift_hom R S φ (a ⊕⇘P⇙ b)) (poly_lift_hom R S φ q)"
using assms UP_cring.sub_add[of R q a b ] UP_cring.sub_add[of S ]
unfolding UP_cring_def
by (metis A(1) A(2) A(3) A(4) P_def R_cring UP_cring.sub_closed UP_cring_axioms poly_lift_hom_add poly_lift_hom_closed)
qed
show "⋀a n. a ∈ carrier R ⟹
poly_lift_hom R S φ (Cring_Poly.compose R (monom P a n) q) =
Cring_Poly.compose S (poly_lift_hom R S φ (monom P a n)) (poly_lift_hom R S φ q)"
proof-
fix a n assume A: "a ∈ carrier R"
have 0: "(poly_lift_hom R S φ (monom (UP R) a n)) = monom (UP S) (φ a) n"
by (simp add: A assms(1) assms(2) assms(3) assms(4) poly_lift_hom_monom)
have 1: " q [^]⇘UP R⇙ n ∈ carrier (UP R)"
using monoid.nat_pow_closed[of "UP R" q n] UP_ring.UP_ring UP_ring.intro assms(1) assms
P.monoid_axioms P_def by blast
have 2: "poly_lift_hom R S φ (to_polynomial R a ⊗⇘UP R⇙ q [^]⇘UP R⇙ n) =
to_polynomial S (φ a) ⊗⇘UP S⇙ (poly_lift_hom R S φ q) [^]⇘UP S⇙ n"
using poly_lift_hom_mult[of S φ "to_polynomial R a" "q [^]⇘UP R⇙ n"] poly_lift_hom_is_hom[of S φ]
ring_hom_nat_pow[of P "UP S" "poly_lift_hom R S φ" q n] UP_cring.UP_cring[of S]
UP_cring poly_lift_hom_monom[of S φ a 0] ring_hom_closed[of φ R S a]
monom_closed[of a 0] nat_pow_closed[of q n] assms A
unfolding to_polynomial_def P_def UP_cring_def cring_def
by auto
have 3: "poly_lift_hom R S φ (Cring_Poly.compose R (monom (UP R) a n) q) = to_polynomial S (φ a) ⊗⇘UP S⇙ (poly_lift_hom R S φ q) [^]⇘UP S⇙ n"
using "2" A P_def assms(4) sub_monom(1) by auto
have 4: "Cring_Poly.compose S (poly_lift_hom R S φ (monom (UP R) a n)) (poly_lift_hom R S φ q)
= Cring_Poly.compose S (monom (UP S) (φ a) n) (poly_lift_hom R S φ q)"
by (simp add: "0")
have "poly_lift_hom R S φ q ∈ carrier (UP S)"
using P_def UP_cring.poly_lift_hom_closed UP_cring_axioms assms(1) assms(2) assms(4) by blast
then have 5: "Cring_Poly.compose S (poly_lift_hom R S φ (monom (UP R) a n)) (poly_lift_hom R S φ q)
= to_polynomial S (φ a) ⊗⇘UP S⇙ (poly_lift_hom R S φ q) [^]⇘UP S⇙ n"
using 4 UP_cring.sub_monom[of S "poly_lift_hom R S φ q" "φ a" n] assms
unfolding UP_cring_def
by (simp add: A ring_hom_closed)
thus "poly_lift_hom R S φ (Cring_Poly.compose R (monom P a n) q) =
Cring_Poly.compose S (poly_lift_hom R S φ (monom P a n)) (poly_lift_hom R S φ q)"
using 0 1 2 3 4 assms A
by (simp add: P_def)
qed
qed
lemma(in UP_cring) poly_lift_hom_comm_taylor_expansion:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
shows "poly_lift_hom R S φ (taylor_expansion R a p) = taylor_expansion S (φ a) (poly_lift_hom R S φ p)"
unfolding taylor_expansion_def
using poly_lift_hom_sub[of S φ p "(X_poly_plus R a)"] poly_lift_hom_X_plus[of S φ a] assms
by (simp add: P_def UP_cring.X_plus_closed UP_cring_axioms)
lemma(in UP_cring) poly_lift_hom_comm_taylor_expansion_cf:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "p ∈ carrier (UP R)"
assumes "a ∈ carrier R"
shows "φ (taylor_expansion R a p i) = taylor_expansion S (φ a) (poly_lift_hom R S φ p) i"
using poly_lift_hom_cf assms poly_lift_hom_comm_taylor_expansion P_def
taylor_def UP_cring.taylor_closed UP_cring_axioms by fastforce
lemma(in UP_cring) taylor_expansion_cf_closed:
assumes "p ∈ carrier P"
assumes "a ∈ carrier R"
shows "taylor_expansion R a p i ∈ carrier R"
using assms taylor_closed
by (simp add: taylor_def cfs_closed)
lemma(in UP_cring) poly_lift_hom_comm_taylor_term:
assumes "cring S"
assumes "φ ∈ ring_hom R S"
assumes "p ∈ carrier (UP R)"
assumes "a ∈ carrier R"
shows "poly_lift_hom R S φ (taylor_term a p i) = UP_cring.taylor_term S (φ a) (poly_lift_hom R S φ p) i"
using poly_lift_hom_X_minus_nat_pow_smult[of S φ a "taylor_expansion R a p i" i]
poly_lift_hom_comm_taylor_expansion[of S φ p a]
poly_lift_hom_comm_taylor_expansion_cf[of S φ p a i]
assms UP_cring.taylor_term_def[of S]
unfolding taylor_term_def UP_cring_def P_def
by (simp add: UP_cring.taylor_expansion_cf_closed UP_cring_axioms)
lemma(in UP_cring) poly_lift_hom_degree_bound:
assumes "cring S"
assumes "h ∈ ring_hom R S"
assumes "f ∈ carrier (UP R)"
shows "deg S (poly_lift_hom R S h f) ≤ deg R f"
using poly_lift_hom_closed[of S h f] UP_cring.deg_leqI[of S "poly_lift_hom R S h f" "deg R f"] assms ring_hom_zero[of h R S] deg_aboveD[of f] coeff_simp[of f]
unfolding P_def UP_cring_def
by (simp add: P_def R.ring_axioms cring.axioms(1) poly_lift_hom_cf)
lemma(in UP_cring) deg_eqI:
assumes "f ∈ carrier (UP R)"
assumes "deg R f ≤ n"
assumes "f n ≠ 𝟬"
shows "deg R f = n"
using assms coeff_simp[of f] P_def deg_leE le_neq_implies_less by blast
lemma(in UP_cring) poly_lift_hom_degree_eq:
assumes "cring S"
assumes "h ∈ ring_hom R S"
assumes "h (lcf f) ≠ 𝟬⇘S⇙"
assumes "f ∈ carrier (UP R)"
shows "deg S (poly_lift_hom R S h f) = deg R f"
apply(rule UP_cring.deg_eqI)
using assms unfolding UP_cring_def apply blast
using poly_lift_hom_closed[of S h f] assms apply blast
using poly_lift_hom_degree_bound[of S h f] assms apply blast
using assms poly_lift_hom_cf[of S h f]
by (metis P_def)
lemma(in UP_cring) poly_lift_hom_lcoeff:
assumes "cring S"
assumes "h ∈ ring_hom R S"
assumes "h (lcf f) ≠ 𝟬⇘S⇙"
assumes "f ∈ carrier (UP R)"
shows "UP_ring.lcf S (poly_lift_hom R S h f) = h (lcf f)"
using poly_lift_hom_degree_eq[of S h f] assms
by (simp add: P_def poly_lift_hom_cf)
end
section‹Coefficient List Constructor for Polynomials›
definition list_to_poly where
"list_to_poly R as n = (if n < length as then as!n else 𝟬⇘R⇙)"
context UP_ring
begin
lemma(in UP_ring) list_to_poly_closed:
assumes "set as ⊆ carrier R"
shows "list_to_poly R as ∈ carrier P"
apply(rule UP_car_memI[of "length as"])
apply (simp add: list_to_poly_def)
by (metis R.zero_closed assms in_mono list_to_poly_def nth_mem)
lemma(in UP_ring) list_to_poly_zero[simp]:
"list_to_poly R [] = 𝟬⇘UP R⇙"
unfolding list_to_poly_def
apply auto
by(simp add: UP_def)
lemma(in UP_domain) list_to_poly_singleton:
assumes "a ∈ carrier R"
shows "list_to_poly R [a] = monom P a 0"
apply(rule ext)
unfolding list_to_poly_def using assms
by (simp add: cfs_monom)
end
definition cf_list where
"cf_list R p = map p [(0::nat)..< Suc (deg R p)]"
lemma cf_list_length:
"length (cf_list R p) = Suc (deg R p)"
unfolding cf_list_def
by simp
lemma cf_list_entries:
assumes "i ≤ deg R p"
shows "(cf_list R p)!i = p i"
unfolding cf_list_def
by (metis add.left_neutral assms diff_zero less_Suc_eq_le map_eq_map_tailrec nth_map_upt)
lemma(in UP_ring) list_to_poly_cf_list_inv:
assumes "p ∈ carrier (UP R)"
shows "list_to_poly R (cf_list R p) = p"
proof
fix x
show "list_to_poly R (cf_list R p) x = p x"
apply(cases "x < degree p")
unfolding list_to_poly_def
using assms cf_list_length[of R p] cf_list_entries[of _ R p]
apply simp
by (metis P_def UP_ring.coeff_simp UP_ring_axioms ‹⋀i. i ≤ deg R p ⟹ cf_list R p ! i = p i› ‹length (cf_list R p) = Suc (deg R p)› assms deg_belowI less_Suc_eq_le)
qed
section‹Polynomial Rings over a Subring›
subsection‹Characterizing the Carrier of a Polynomial Ring over a Subring›
lemma(in ring) carrier_update:
"carrier (R⦇carrier := S⦈) = S"
"𝟬⇘(R⦇carrier := S⦈)⇙ = 𝟬"
"𝟭⇘(R⦇carrier := S⦈)⇙ = 𝟭"
"(⊕⇘(R⦇carrier := S⦈)⇙) = (⊕)"
"(⊗⇘(R⦇carrier := S⦈)⇙) = (⊗)"
by auto
lemma(in UP_cring) poly_cfs_subring:
assumes "subring S R"
assumes "g ∈ carrier (UP R)"
assumes "⋀n. g n ∈ S"
shows "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
apply(rule UP_cring.UP_car_memI')
using R.subcringI' R.subcring_iff UP_cring.intro assms(1) subringE(1) apply blast
proof-
have "carrier (R⦇carrier := S⦈) = S"
using ring.carrier_update by simp
then show 0: "⋀x. g x ∈ carrier (R⦇carrier := S⦈)"
using assms by blast
have 0: "𝟬⇘R⦇carrier := S⦈⇙ = 𝟬"
using R.carrier_update(2) by blast
then show "⋀x. (deg R g) < x ⟹ g x = 𝟬⇘R⦇carrier := S⦈⇙"
using UP_car_memE assms(2) by presburger
qed
lemma(in UP_cring) UP_ring_subring:
assumes "subring S R"
shows "UP_cring (R ⦇ carrier := S ⦈)" "UP_ring (R ⦇ carrier := S ⦈)"
using assms unfolding UP_cring_def
using R.subcringI' R.subcring_iff subringE(1) apply blast
using assms unfolding UP_ring_def
using R.subcringI' R.subcring_iff subringE(1)
by (simp add: R.subring_is_ring)
lemma(in UP_cring) UP_ring_subring_is_ring:
assumes "subring S R"
shows "cring (UP (R ⦇ carrier := S ⦈))"
using assms UP_ring_subring[of S] UP_cring.UP_cring[of "R⦇carrier := S⦈"]
by blast
lemma(in UP_cring) UP_ring_subring_add_closed:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "f ∈ carrier (UP (R ⦇ carrier := S ⦈))"
shows "f ⊕⇘UP (R ⦇ carrier := S ⦈)⇙g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
using assms UP_ring_subring_is_ring[of S]
by (meson cring.cring_simprules(1))
lemma(in UP_cring) UP_ring_subring_mult_closed:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "f ∈ carrier (UP (R ⦇ carrier := S ⦈))"
shows "f ⊗⇘UP (R ⦇ carrier := S ⦈)⇙g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
using assms UP_ring_subring_is_ring[of S]
by (meson cring.carrier_is_subcring subcringE(6))
lemma(in UP_cring) UP_ring_subring_car:
assumes "subring S R"
shows "carrier (UP (R ⦇ carrier := S ⦈)) = {h ∈ carrier (UP R). ∀n. h n ∈ S}"
proof
show "carrier (UP (R⦇carrier := S⦈)) ⊆ {h ∈ carrier (UP R). ∀n. h n ∈ S}"
proof
fix h assume A: "h ∈ carrier (UP (R⦇carrier := S⦈))"
have "h ∈ carrier P"
apply(rule UP_car_memI[of "deg (R⦇carrier := S⦈) h"]) unfolding P_def
using UP_cring.UP_car_memE[of "R⦇carrier := S⦈" h] R.carrier_update[of S]
assms UP_ring_subring A apply presburger
using UP_cring.UP_car_memE[of "R⦇carrier := S⦈" h] assms
by (metis A R.ring_axioms UP_cring_def ‹carrier (R⦇carrier := S⦈) = S› cring.subcringI' is_UP_cring ring.subcring_iff subringE(1) subsetD)
then show "h ∈ {h ∈ carrier (UP R). ∀n. h n ∈ S}"
unfolding P_def
using assms A UP_cring.UP_car_memE[of "R⦇carrier := S⦈" h] R.carrier_update[of S]
UP_ring_subring by blast
qed
show "{h ∈ carrier (UP R). ∀n. h n ∈ S} ⊆ carrier (UP (R⦇carrier := S⦈))"
proof fix h assume A: "h ∈ {h ∈ carrier (UP R). ∀n. h n ∈ S}"
have 0: "h ∈ carrier (UP R)"
using A by blast
have 1: "⋀n. h n ∈ S"
using A by blast
show "h ∈ carrier (UP (R⦇carrier := S⦈))"
apply(rule UP_ring.UP_car_memI[of _ "deg R h"])
using assms UP_ring_subring[of S] UP_cring.axioms UP_ring.intro cring.axioms(1) apply blast
using UP_car_memE[of h] carrier_update 0 R.carrier_update(2) apply presburger
using assms 1 R.carrier_update(1) by blast
qed
qed
lemma(in UP_cring) UP_ring_subring_car_subset:
assumes "subring S R"
shows "carrier (UP (R ⦇ carrier := S ⦈)) ⊆ carrier (UP R)"
proof fix h assume "h ∈ carrier (UP (R ⦇ carrier := S ⦈))"
then show "h ∈ carrier (UP R)"
using assms UP_ring_subring_car[of S] by blast
qed
lemma(in UP_cring) UP_ring_subring_car_subset':
assumes "subring S R"
assumes "h ∈ carrier (UP (R ⦇ carrier := S ⦈))"
shows "h ∈ carrier (UP R)"
using assms UP_ring_subring_car_subset[of S] by blast
lemma(in UP_cring) UP_ring_subring_add:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "f ∈ carrier (UP (R ⦇ carrier := S ⦈))"
shows "g ⊕⇘UP R⇙ f = g ⊕⇘UP (R ⦇ carrier := S ⦈)⇙f"
proof(rule ext) fix x show "(g ⊕⇘UP R⇙ f) x = (g ⊕⇘UP (R⦇carrier := S⦈)⇙ f) x"
proof-
have 0: " (g ⊕⇘P⇙ f) x = g x ⊕ f x"
using assms cfs_add[of g f x] unfolding P_def
using UP_ring_subring_car_subset' by blast
have 1: "(g ⊕⇘UP (R⦇carrier := S⦈)⇙ f) x = g x ⊕⇘R⦇carrier := S⦈⇙ f x"
using UP_ring.cfs_add[of "R ⦇ carrier := S ⦈" g f x] UP_ring_subring[of S] assms
unfolding UP_ring_def UP_cring_def
using R.subring_is_ring by blast
show ?thesis using 0 1 R.carrier_update(4)[of S]
by (simp add: P_def)
qed
qed
lemma(in UP_cring) UP_ring_subring_deg:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
shows "deg R g = deg (R ⦇ carrier := S ⦈) g"
proof-
have 0: "g ∈ carrier (UP R)"
using assms UP_ring_subring_car[of S] by blast
have 1: "deg R g ≤ deg (R ⦇ carrier := S ⦈) g"
using 0 assms UP_cring.UP_car_memE[of "R ⦇ carrier := S ⦈" g]
UP_car_memE[of g] P_def R.carrier_update(2) UP_ring_subring deg_leqI by presburger
have 2: "deg (R ⦇ carrier := S ⦈) g ≤ deg R g"
using 0 assms UP_cring.UP_car_memE[of "R ⦇ carrier := S ⦈" g]
UP_car_memE[of g] P_def R.carrier_update(2) UP_ring_subring UP_cring.deg_leqI
by metis
show ?thesis using 1 2 by presburger
qed
lemma(in UP_cring) UP_subring_monom:
assumes "subring S R"
assumes "a ∈ S"
shows "up_ring.monom (UP R) a n = up_ring.monom (UP (R ⦇ carrier := S ⦈)) a n"
proof fix x
have 0: "a ∈ carrier R"
using assms subringE(1) by blast
have 1: "a ∈ carrier (R⦇carrier := S⦈)"
using assms by (simp add: assms(2))
have 2: " up_ring.monom (UP (R⦇carrier := S⦈)) a n x = (if n = x then a else 𝟬⇘R⦇carrier := S⦈⇙)"
using 1 assms UP_ring_subring[of S] UP_ring.cfs_monom[of "R⦇carrier := S⦈" a n x] UP_cring.axioms UP_ring.intro cring.axioms(1)
by blast
show "up_ring.monom (UP R) a n x = up_ring.monom (UP (R⦇carrier := S⦈)) a n x"
using 0 1 2 cfs_monom[of a n x] R.carrier_update(2)[of S] unfolding P_def by presburger
qed
lemma(in UP_cring) UP_ring_subring_mult:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "f ∈ carrier (UP (R ⦇ carrier := S ⦈))"
shows "g ⊗⇘UP R⇙ f = g ⊗⇘UP (R ⦇ carrier := S ⦈)⇙f"
proof(rule UP_ring.poly_induct3[of "R ⦇ carrier := S ⦈" f])
show "UP_ring (R⦇carrier := S⦈)"
by (simp add: UP_ring_subring(2) assms(1))
show " f ∈ carrier (UP (R⦇carrier := S⦈))"
by (simp add: assms(3))
show " ⋀p q. q ∈ carrier (UP (R⦇carrier := S⦈)) ⟹
p ∈ carrier (UP (R⦇carrier := S⦈)) ⟹
g ⊗⇘UP R⇙ p = g ⊗⇘UP (R⦇carrier := S⦈)⇙ p ⟹
g ⊗⇘UP R⇙ q = g ⊗⇘UP (R⦇carrier := S⦈)⇙ q ⟹ g ⊗⇘UP R⇙ (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q) = g ⊗⇘UP (R⦇carrier := S⦈)⇙ (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q)"
proof- fix p q
assume A: " q ∈ carrier (UP (R⦇carrier := S⦈))"
"p ∈ carrier (UP (R⦇carrier := S⦈))"
"g ⊗⇘UP R⇙ p = g ⊗⇘UP (R⦇carrier := S⦈)⇙ p"
"g ⊗⇘UP R⇙ q = g ⊗⇘UP (R⦇carrier := S⦈)⇙ q"
have 0: "p ⊕⇘UP (R⦇carrier := S⦈)⇙ q = p ⊕⇘UP R⇙ q"
using A UP_ring_subring_add[of S p q]
by (simp add: assms(1))
have 1: "g ⊗⇘UP R⇙ (p ⊕⇘UP R⇙ q) = g ⊗⇘UP R⇙ p ⊕⇘UP R⇙ g ⊗⇘UP R⇙ q"
using 0 A assms P.r_distr P_def UP_ring_subring_car_subset' by auto
hence 2:"g ⊗⇘UP R⇙ (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q) = g ⊗⇘UP R⇙ p ⊕⇘UP R⇙ g ⊗⇘UP R⇙ q"
using 0 by simp
have 3: "g ⊗⇘UP (R⦇carrier := S⦈)⇙ (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q) =
g ⊗⇘UP (R⦇carrier := S⦈)⇙ p ⊕⇘UP (R⦇carrier := S⦈)⇙ g ⊗⇘UP (R⦇carrier := S⦈)⇙ q"
using 0 A assms semiring.r_distr[of "UP (R⦇carrier := S⦈)"] UP_ring_subring_car_subset'
using UP_ring.UP_r_distr ‹UP_ring (R⦇carrier := S⦈)› by blast
hence 4: "g ⊗⇘UP (R⦇carrier := S⦈)⇙ (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q) =
g ⊗⇘UP R⇙ p ⊕⇘UP (R⦇carrier := S⦈)⇙ g ⊗⇘UP R⇙ q"
using A by simp
hence 5: "g ⊗⇘UP (R⦇carrier := S⦈)⇙ (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q) =
g ⊗⇘UP R⇙ p ⊕⇘UP R⇙ g ⊗⇘UP R⇙ q"
using UP_ring_subring_add[of S]
by (simp add: A(1) A(2) A(3) A(4) UP_ring.UP_mult_closed ‹UP_ring (R⦇carrier := S⦈)› assms(1) assms(2))
show "g ⊗⇘UP R⇙ (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q) = g ⊗⇘UP (R⦇carrier := S⦈)⇙ (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q)"
by (simp add: "2" "5")
qed
show "⋀a n. a ∈ carrier (R⦇carrier := S⦈) ⟹ g ⊗⇘UP R⇙ monom (UP (R⦇carrier := S⦈)) a n = g ⊗⇘UP (R⦇carrier := S⦈)⇙ monom (UP (R⦇carrier := S⦈)) a n"
proof fix a n x assume A: "a ∈ carrier (R⦇carrier := S⦈)"
have 0: "monom (UP (R⦇carrier := S⦈)) a n = monom (UP R) a n"
using A UP_subring_monom assms(1) by auto
have 1: "g ∈ carrier (UP R)"
using assms UP_ring_subring_car_subset' by blast
have 2: "a ∈ carrier R"
using A assms subringE(1)[of S R] R.carrier_update[of S] by blast
show "(g ⊗⇘UP R⇙ monom (UP (R⦇carrier := S⦈)) a n) x = (g ⊗⇘UP (R⦇carrier := S⦈)⇙ monom (UP (R⦇carrier := S⦈)) a n) x"
proof(cases "x < n")
case True
have T0: "(g ⊗⇘UP R⇙ monom (UP R) a n) x = 𝟬"
using 1 2 True cfs_monom_mult[of g a x n] A assms unfolding P_def by blast
then show ?thesis using UP_cring.cfs_monom_mult[of "R⦇carrier := S⦈" g a x n] 0 A True
UP_ring_subring(1) assms(1) assms(2) by auto
next
case False
have F0: "(g ⊗⇘UP R⇙ monom (UP R) a n) x = a ⊗ (g (x - n))"
using 1 2 False cfs_monom_mult_l[of g a n "x - n"] A assms unfolding P_def by simp
have F1: "(g ⊗⇘UP (R⦇carrier := S⦈)⇙ monom (UP (R⦇carrier := S⦈)) a n) (x - n + n) = a ⊗⇘R⦇carrier := S⦈⇙ g (x - n)"
using 1 2 False UP_cring.cfs_monom_mult_l[of "R⦇carrier := S⦈" g a n "x - n"] A assms
UP_ring_subring(1) by blast
hence F2: "(g ⊗⇘UP (R⦇carrier := S⦈)⇙ monom (UP R) a n) (x - n + n) = a ⊗ g (x - n)"
using UP_subring_monom[of S a n] R.carrier_update[of S] assms 0 by metis
show ?thesis using F0 F1 1 2 assms
by (simp add: "0" False add.commute add_diff_inverse_nat)
qed
qed
qed
lemma(in UP_cring) UP_ring_subring_one:
assumes "subring S R"
shows "𝟭⇘UP R⇙ = 𝟭⇘UP (R ⦇ carrier := S ⦈)⇙"
using UP_subring_monom[of S 𝟭 0] assms P_def R.subcringI' UP_ring.monom_one UP_ring_subring(2) monom_one subcringE(3) by force
lemma(in UP_cring) UP_ring_subring_zero:
assumes "subring S R"
shows "𝟬⇘UP R⇙ = 𝟬⇘UP (R ⦇ carrier := S ⦈)⇙"
using UP_subring_monom[of S 𝟬 0] UP_ring.monom_zero[of "R ⦇ carrier := S ⦈" 0] assms monom_zero[of 0]
UP_ring_subring[of S] subringE(2)[of S R]
unfolding P_def
by (simp add: P_def R.carrier_update(2))
lemma(in UP_cring) UP_ring_subring_nat_pow:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
shows "g[^]⇘UP R⇙n = g[^]⇘UP (R ⦇ carrier := S ⦈)⇙(n::nat)"
apply(induction n)
using assms apply (simp add: UP_ring_subring_one)
proof-
fix n::nat
assume A: "g [^]⇘UP R⇙ n = g [^]⇘UP (R⦇carrier := S⦈)⇙ n"
have "Group.monoid (UP (R⦇carrier := S⦈)) "
using assms UP_ring_subring[of S] UP_ring.UP_ring[of "R⦇carrier := S⦈"] ring.is_monoid by blast
hence 0 : " g [^]⇘UP (R⦇carrier := S⦈)⇙ n ∈ carrier (UP (R⦇carrier := S⦈))"
using monoid.nat_pow_closed[of "UP (R ⦇ carrier := S ⦈)" g n] assms UP_ring_subring
unfolding UP_ring_def ring_def by blast
have 1: "g [^]⇘UP R⇙ n ∈ carrier (UP R)"
using 0 assms UP_ring_subring_car_subset'[of S] by (simp add: A)
then have 2: "g [^]⇘UP R⇙ n ⊗⇘UP R⇙ g = g [^]⇘UP (R⦇carrier := S⦈)⇙ n ⊗⇘UP (R⦇carrier := S⦈)⇙ g"
using assms UP_ring_subring_mult[of S "g [^]⇘UP R⇙ n" g]
by (simp add: "0" A)
then show "g [^]⇘UP R⇙ Suc n = g [^]⇘UP (R⦇carrier := S⦈)⇙ Suc n"
by simp
qed
lemma(in UP_cring) UP_subring_compose_monom:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "a ∈ S"
shows "compose R (up_ring.monom (UP R) a n) g = compose (R ⦇ carrier := S ⦈) (up_ring.monom (UP (R ⦇ carrier := S ⦈)) a n) g"
proof-
have g_closed: "g ∈ carrier (UP R)"
using assms UP_ring_subring_car by blast
have 0: "a ∈ carrier R"
using assms subringE(1) by blast
have 1: "compose R (up_ring.monom (UP R) a n) g = a ⊙⇘UP R⇙ (g[^]⇘UP R⇙n)"
using monom_sub[of a g n] unfolding P_def
using "0" assms(2) g_closed by blast
have 2: "compose (R⦇carrier := S⦈) (up_ring.monom (UP (R⦇carrier := S⦈)) a n) g = a ⊙⇘UP (R⦇carrier := S⦈)⇙ g [^]⇘UP (R⦇carrier := S⦈)⇙ n"
using assms UP_cring.monom_sub[of "R ⦇ carrier := S ⦈" a g n] UP_ring_subring[of S] R.carrier_update[of S]
by blast
have 3: " g [^]⇘UP (R⦇carrier := S⦈)⇙ n = g[^]⇘UP R⇙n"
using UP_ring_subring_nat_pow[of S g n]
by (simp add: assms(1) assms(2))
have 4: "a ⊙⇘UP R⇙ (g[^]⇘UP R⇙n) = a ⊙⇘UP (R⦇carrier := S⦈)⇙ g [^]⇘UP (R⦇carrier := S⦈)⇙ n"
proof fix x
show "(a ⊙⇘UP R⇙ g [^]⇘UP R⇙ n) x = (a ⊙⇘UP (R⦇carrier := S⦈)⇙ g [^]⇘UP (R⦇carrier := S⦈)⇙ n) x"
proof-
have LHS: "(a ⊙⇘UP R⇙ g [^]⇘UP R⇙ n) x = a ⊗ ((g [^]⇘UP R⇙ n) x)"
using "0" P.nat_pow_closed P_def cfs_smult g_closed by auto
have RHS: "(a ⊙⇘UP (R⦇carrier := S⦈)⇙ g [^]⇘UP (R⦇carrier := S⦈)⇙ n) x = a ⊗⇘R⦇carrier := S⦈⇙ ((g [^]⇘UP (R⦇carrier := S⦈)⇙ n) x)"
proof-
have "Group.monoid (UP (R⦇carrier := S⦈)) "
using assms UP_ring_subring[of S] UP_ring.UP_ring[of "R⦇carrier := S⦈"] ring.is_monoid by blast
hence 0 : " g [^]⇘UP (R⦇carrier := S⦈)⇙ n ∈ carrier (UP (R⦇carrier := S⦈))"
using monoid.nat_pow_closed[of "UP (R ⦇ carrier := S ⦈)" g n] assms UP_ring_subring
unfolding UP_ring_def ring_def by blast
have 1: "g [^]⇘UP (R⦇carrier := S⦈)⇙ n ∈ carrier (UP (R⦇carrier := S⦈))"
using assms UP_ring_subring[of S] R.carrier_update[of S] 0 by blast
then show ?thesis using UP_ring.cfs_smult UP_ring_subring assms
by (simp add: UP_ring.cfs_smult)
qed
show ?thesis using R.carrier_update RHS LHS 3 assms
by simp
qed
qed
show ?thesis using 0 1 2 3 4
by simp
qed
lemma(in UP_cring) UP_subring_compose:
assumes "subring S R"
assumes "g ∈ carrier (UP R)"
assumes "f ∈ carrier (UP R)"
assumes "⋀n. g n ∈ S"
assumes "⋀n. f n ∈ S"
shows "compose R f g = compose (R ⦇ carrier := S ⦈) f g"
proof-
have g_closed: "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
using assms poly_cfs_subring by blast
have 0: "⋀n. (∀ h. h ∈ carrier (UP R) ∧ deg R h ≤ n ∧ h ∈ carrier (UP (R ⦇ carrier := S ⦈)) ⟶ compose R h g = compose (R ⦇ carrier := S ⦈) h g)"
proof- fix n show "(∀ h. h ∈ carrier (UP R) ∧ deg R h ≤ n ∧ h ∈ carrier (UP (R ⦇ carrier := S ⦈)) ⟶ compose R h g = compose (R ⦇ carrier := S ⦈) h g)"
proof(induction n)
show "∀h. h ∈ carrier (UP R) ∧ deg R h ≤ 0 ∧ h ∈ carrier (UP (R⦇carrier := S⦈)) ⟶ Cring_Poly.compose R h g = Cring_Poly.compose (R⦇carrier := S⦈) h g"
proof fix h
show "h ∈ carrier (UP R) ∧ deg R h ≤ 0 ∧ h ∈ carrier (UP (R⦇carrier := S⦈)) ⟶ Cring_Poly.compose R h g = Cring_Poly.compose (R⦇carrier := S⦈) h g"
proof
assume A: "h ∈ carrier (UP R) ∧ deg R h ≤ 0 ∧ h ∈ carrier (UP (R⦇carrier := S⦈))"
then have 0: "deg R h = 0"
by linarith
then have 1: "deg (R ⦇ carrier := S ⦈) h = 0"
using A assms UP_ring_subring_deg[of S h]
by linarith
show "Cring_Poly.compose R h g = Cring_Poly.compose (R⦇carrier := S⦈) h g"
using 0 1 g_closed assms sub_const[of g h] UP_cring.sub_const[of "R⦇carrier := S⦈" g h] A P_def UP_ring_subring
by presburger
qed
qed
show "⋀n. ∀h. h ∈ carrier (UP R) ∧ deg R h ≤ n ∧ h ∈ carrier (UP (R⦇carrier := S⦈)) ⟶
Cring_Poly.compose R h g = Cring_Poly.compose (R⦇carrier := S⦈) h g ⟹
∀h. h ∈ carrier (UP R) ∧ deg R h ≤ Suc n ∧ h ∈ carrier (UP (R⦇carrier := S⦈)) ⟶
Cring_Poly.compose R h g = Cring_Poly.compose (R⦇carrier := S⦈) h g"
proof fix n h
assume IH: "∀h. h ∈ carrier (UP R) ∧ deg R h ≤ n ∧ h ∈ carrier (UP (R⦇carrier := S⦈)) ⟶
Cring_Poly.compose R h g = Cring_Poly.compose (R⦇carrier := S⦈) h g"
show "h ∈ carrier (UP R) ∧ deg R h ≤ Suc n ∧ h ∈ carrier (UP (R⦇carrier := S⦈)) ⟶
Cring_Poly.compose R h g = Cring_Poly.compose (R⦇carrier := S⦈) h g"
proof assume A: "h ∈ carrier (UP R) ∧ deg R h ≤ Suc n ∧ h ∈ carrier (UP (R⦇carrier := S⦈))"
show "Cring_Poly.compose R h g = Cring_Poly.compose (R⦇carrier := S⦈) h g"
proof(cases "deg R h ≤ n")
case True
then show ?thesis using A IH by blast
next
case False
then have F0: "deg R h = Suc n"
using A by (simp add: A le_Suc_eq)
then have F1: "deg (R⦇carrier := S⦈) h = Suc n"
using UP_ring_subring_deg[of S h] A
by (simp add: ‹h ∈ carrier (UP R) ∧ deg R h ≤ Suc n ∧ h ∈ carrier (UP (R⦇carrier := S⦈))› assms(1))
obtain j where j_def: "j ∈ carrier (UP (R⦇carrier := S⦈)) ∧
h = j ⊕⇘UP (R⦇carrier := S⦈)⇙ up_ring.monom (UP (R⦇carrier := S⦈)) (h (deg (R⦇carrier := S⦈) h)) (deg (R⦇carrier := S⦈) h) ∧
deg (R⦇carrier := S⦈) j < deg (R⦇carrier := S⦈) h"
using A UP_ring.ltrm_decomp[of "R⦇carrier := S⦈" h] assms UP_ring_subring[of S]
F1 by (metis (mono_tags, lifting) F0 False zero_less_Suc)
have j_closed: "j ∈ carrier (UP R)"
using j_def assms UP_ring_subring_car_subset by blast
have F2: "deg R j < deg R h"
using j_def assms
by (metis (no_types, lifting) F0 F1 UP_ring_subring_deg)
have F3: "(deg (R⦇carrier := S⦈) h) = deg R h"
by (simp add: F0 F1)
have F30: "h (deg (R⦇carrier := S⦈) h) ∈ S "
using A UP_cring.UP_car_memE[of "R⦇carrier := S⦈" h "deg (R⦇carrier := S⦈) h"]
by (metis R.carrier_update(1) UP_ring_subring(1) assms(1))
hence F4: "up_ring.monom P (h (deg R h)) (deg R h) =
up_ring.monom (UP (R⦇carrier := S⦈)) (h (deg (R⦇carrier := S⦈) h)) (deg (R⦇carrier := S⦈) h)"
using F3 g_closed j_def UP_subring_monom[of S "h (deg (R⦇carrier := S⦈) h)"] assms
unfolding P_def by metis
have F5: "compose R (up_ring.monom (UP R) (h (deg R h)) (deg R h)) g =
compose (R ⦇ carrier := S ⦈) (up_ring.monom (UP (R ⦇ carrier := S ⦈)) (h (deg (R⦇carrier := S⦈) h)) (deg (R⦇carrier := S⦈) h)) g"
using F0 F1 F2 F3 F4 UP_subring_compose_monom[of S] assms P_def ‹h (deg (R⦇carrier := S⦈) h) ∈ S›
by (metis g_closed)
have F5: "compose R j g = compose (R ⦇ carrier := S ⦈) j g"
using F0 F2 IH UP_ring_subring_car_subset' assms(1) j_def by auto
have F6: "h = j ⊕⇘UP R⇙ monom (UP R) (h (deg R h)) (deg R h)"
using j_def F4 UP_ring_subring_add[of S j "up_ring.monom (UP (R⦇carrier := S⦈)) (h (deg (R⦇carrier := S⦈) h)) (deg (R⦇carrier := S⦈) h)"]
UP_ring.monom_closed[of "R⦇carrier := S⦈" "h (deg (R⦇carrier := S⦈) h)" "deg (R⦇carrier := S⦈) h"]
using P_def UP_ring_subring(2) ‹h (deg (R⦇carrier := S⦈) h) ∈ S› assms(1) by auto
have F7: "compose R h g =compose R j g ⊕⇘UP R⇙
compose R (up_ring.monom (UP R) (h (deg R h)) (deg R h)) g"
proof-
show ?thesis
using assms(2) j_closed F5 sub_add[of g j "up_ring.monom P (h (deg R h)) (deg R h)" ]
F4 F3 F2 F1 g_closed unfolding P_def
by (metis A F6 ltrm_closed P_def)
qed
have F8: "compose (R ⦇ carrier := S ⦈) h g = compose (R ⦇ carrier := S ⦈) j g ⊕⇘UP (R ⦇ carrier := S ⦈)⇙
compose (R ⦇ carrier := S ⦈) (up_ring.monom (UP (R ⦇ carrier := S ⦈)) (h (deg (R ⦇ carrier := S ⦈) h)) (deg (R ⦇ carrier := S ⦈) h)) g"
proof-
have 0: " UP_cring (R⦇carrier := S⦈)"
by (simp add: UP_ring_subring(1) assms(1))
have 1: "monom (UP (R⦇carrier := S⦈)) (h (deg R h)) (deg R h) ∈ carrier (UP (R⦇carrier := S⦈))"
using assms 0 F30 UP_ring.monom_closed[of "R⦇carrier := S⦈" "h (deg R h)" "deg R h"] R.carrier_update[of S]
unfolding UP_ring_def UP_cring_def
by (simp add: F3 cring.axioms(1))
show ?thesis
using 0 1 g_closed j_def UP_cring.sub_add[of "R ⦇ carrier := S ⦈" g j "monom (UP (R⦇carrier := S⦈)) (h (deg R h)) (deg R h)" ]
using F3 by auto
qed
have F9: "compose R j g ∈ carrier (UP R)"
by (simp add: UP_cring.sub_closed assms(2) is_UP_cring j_closed)
have F10: "compose (R ⦇ carrier := S ⦈) j g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
using assms j_def UP_cring.sub_closed[of "R ⦇ carrier := S ⦈"] UP_ring_subring(1) g_closed by blast
have F11: " compose R (up_ring.monom (UP R) (h (deg R h)) (deg R h)) g ∈ carrier (UP R)"
using assms j_def UP_cring.sub_closed[of "R ⦇ carrier := S ⦈"]
UP_ring.monom_closed[of "R ⦇ carrier := S ⦈"]
by (simp add: A UP_car_memE(1) UP_cring.rev_sub_closed UP_ring.monom_closed is_UP_cring is_UP_ring sub_rev_sub)
have F12: " compose (R ⦇ carrier := S ⦈) (up_ring.monom (UP (R ⦇ carrier := S ⦈)) (h (deg (R ⦇ carrier := S ⦈) h)) (deg (R ⦇ carrier := S ⦈) h)) g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
using assms j_def UP_cring.sub_closed[of "R ⦇ carrier := S ⦈"]
UP_ring.monom_closed[of "R ⦇ carrier := S ⦈"] UP_ring_subring[of S]
using A UP_ring.ltrm_closed g_closed by fastforce
show ?thesis using F9 F10 F11 F12 F7 F8 F5 UP_ring_subring_add[of S "compose R j g" "compose R (up_ring.monom (UP R) (h (deg R h)) (deg R h)) g"]
assms
using F3 F30 UP_subring_compose_monom g_closed by auto
qed
qed
qed
qed
qed
show ?thesis using 0[of "deg R f"]
by (simp add: assms(1) assms(3) assms(5) poly_cfs_subring)
qed
subsection‹Evaluation over a Subring›
lemma(in UP_cring) UP_subring_eval:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "a ∈ S"
shows "to_function R g a = to_function (R ⦇ carrier := S ⦈) g a"
apply(rule UP_ring.poly_induct3[of "R ⦇ carrier := S ⦈" g] )
apply (simp add: UP_ring_subring(2) assms(1))
apply (simp add: assms(2))
proof-
show "⋀p q. q ∈ carrier (UP (R⦇carrier := S⦈)) ⟹
p ∈ carrier (UP (R⦇carrier := S⦈)) ⟹
to_function R p a = to_function (R⦇carrier := S⦈) p a ⟹
to_function R q a = to_function (R⦇carrier := S⦈) q a ⟹
to_function R (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q) a = to_function (R⦇carrier := S⦈) (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q) a"
proof- fix p q assume A: "q ∈ carrier (UP (R⦇carrier := S⦈))"
"p ∈ carrier (UP (R⦇carrier := S⦈))"
" to_function R p a = to_function (R⦇carrier := S⦈) p a"
" to_function R q a = to_function (R⦇carrier := S⦈) q a"
have a_closed: "a ∈ carrier R"
using assms R.carrier_update[of S] subringE(1) by blast
have 0: "UP_cring (R⦇carrier := S⦈)"
using assms by (simp add: UP_ring_subring(1))
have 1: "to_function (R⦇carrier := S⦈) p a ∈ S"
using A 0 UP_cring.to_fun_closed[of "R⦇carrier := S⦈"]
by (simp add: UP_cring.to_fun_def assms(3))
have 2: "to_function (R⦇carrier := S⦈) q a ∈ S"
using A 0 UP_cring.to_fun_closed[of "R⦇carrier := S⦈"]
by (simp add: UP_cring.to_fun_def assms(3))
have 3: "p ∈ carrier (UP R)"
using A assms 0 UP_ring_subring_car_subset' by blast
have 4: "q ∈ carrier (UP R)"
using A assms 0 UP_ring_subring_car_subset' by blast
have 5: "to_fun p a ⊕ to_fun q a = UP_cring.to_fun (R⦇carrier := S⦈) p a ⊕⇘R⦇carrier := S⦈⇙ UP_cring.to_fun (R⦇carrier := S⦈) q a"
using 1 2 A R.carrier_update[of S] assms by (simp add: "0" UP_cring.to_fun_def to_fun_def)
have 6: "UP_cring.to_fun (R⦇carrier := S⦈) (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q) a =
UP_cring.to_fun (R⦇carrier := S⦈) p a ⊕⇘R⦇carrier := S⦈⇙ UP_cring.to_fun (R⦇carrier := S⦈) q a"
using UP_cring.to_fun_plus[of "R ⦇ carrier := S ⦈" q p a]
by (simp add: "0" A(1) A(2) assms(3))
have 7: "to_fun (p ⊕⇘P⇙ q) a = to_fun p a ⊕ to_fun q a"
using to_fun_plus[of q p a] 3 4 a_closed by (simp add: P_def)
have 8: "p ⊕⇘UP (R⦇carrier := S⦈)⇙ q = p ⊕⇘P⇙ q"
unfolding P_def using assms A R.carrier_update[of S] UP_ring_subring_add[of S p q] by simp
show "to_function R (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q) a = to_function (R⦇carrier := S⦈) (p ⊕⇘UP (R⦇carrier := S⦈)⇙ q) a"
using UP_ring_subring_car_subset'[of S ] 0 1 2 3 4 5 6 7 8 A R.carrier_update[of S]
unfolding P_def by (simp add: UP_cring.to_fun_def to_fun_def)
qed
show "⋀b n.
b ∈ carrier (R⦇carrier := S⦈) ⟹
to_function R (monom (UP (R⦇carrier := S⦈)) b n) a = to_function (R⦇carrier := S⦈) (monom (UP (R⦇carrier := S⦈)) b n) a"
proof- fix b n assume A: "b ∈ carrier (R⦇carrier := S⦈)"
have 0: "UP_cring (R⦇carrier := S⦈)"
by (simp add: UP_ring_subring(1) assms(1))
have a_closed: "a ∈ carrier R"
using assms subringE by blast
have 1: "UP_cring.to_fun (R⦇carrier := S⦈) (monom (UP (R⦇carrier := S⦈)) b n) a = b ⊗⇘R⦇carrier := S⦈⇙ a [^]⇘R⦇carrier := S⦈⇙ n"
using assms A UP_cring.to_fun_monom[of "R⦇carrier := S⦈" b a n]
by (simp add: "0")
have 2: "UP_cring.to_fun (R⦇carrier := S⦈) (monom (UP (R⦇carrier := S⦈)) b n) ≡ to_function (R⦇carrier := S⦈) (monom (UP (R⦇carrier := S⦈)) b n)"
using UP_cring.to_fun_def[of "R⦇carrier := S⦈" "monom (UP (R⦇carrier := S⦈)) b n"] 0 by linarith
have 3: "(monom (UP (R⦇carrier := S⦈)) b n) = monom P b n"
using A assms unfolding P_def using UP_subring_monom by auto
have 4: " b ⊗ a [^] n = b ⊗⇘R⦇carrier := S⦈⇙ a [^]⇘R⦇carrier := S⦈⇙ n"
apply(induction n) using R.carrier_update[of S] apply simp
using R.carrier_update[of S] R.nat_pow_consistent by auto
hence 5: "to_function R (monom (UP (R⦇carrier := S⦈)) b n) a = b ⊗⇘R⦇carrier := S⦈⇙ a[^]⇘R⦇carrier := S⦈⇙n"
using 0 1 2 3 assms A UP_cring.to_fun_monom[of "R⦇carrier := S⦈" b a n] UP_cring.to_fun_def[of "R⦇carrier := S⦈" "monom (UP (R⦇carrier := S⦈)) b n"]
R.carrier_update[of S] subringE[of S R] a_closed UP_ring.monom_closed[of "R⦇carrier := S⦈" a n]
to_fun_monom[of b a n]
unfolding P_def UP_cring.to_fun_def to_fun_def by (metis subsetD)
thus " to_function R (monom (UP (R⦇carrier := S⦈)) b n) a = to_function (R⦇carrier := S⦈) (monom (UP (R⦇carrier := S⦈)) b n) a"
using "1" "2" by auto
qed
qed
lemma(in UP_cring) UP_subring_eval':
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "a ∈ S"
shows "to_fun g a = to_function (R ⦇ carrier := S ⦈) g a"
unfolding to_fun_def using assms
by (simp add: UP_subring_eval)
lemma(in UP_cring) UP_subring_eval_closed:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "a ∈ S"
shows "to_fun g a ∈ S"
using assms UP_subring_eval'[of S g a] UP_cring.to_fun_closed UP_cring.to_fun_def R.carrier_update(1) UP_ring_subring(1) by fastforce
subsection‹Derivatives and Taylor Expansions over a Subring›
lemma(in UP_cring) UP_subring_taylor:
assumes "subring S R"
assumes "g ∈ carrier (UP R)"
assumes "⋀n. g n ∈ S"
assumes "a ∈ S"
shows "taylor_expansion R a g = taylor_expansion (R ⦇ carrier := S ⦈) a g"
proof-
have a_closed: "a ∈ carrier R"
using assms subringE by blast
have 0: "X_plus a ∈ carrier (UP R)"
using assms X_plus_closed unfolding P_def
using local.a_closed by auto
have 1: "⋀n. X_plus a n ∈ S"
proof- fix n
have "X_plus a n = (if n = 0 then a else
(if n = 1 then 𝟭 else 𝟬))"
using a_closed
by (simp add: cfs_X_plus)
then show "X_plus a n ∈ S" using subringE assms
by (simp add: subringE(2) subringE(3))
qed
have 2: "(X_poly_plus (R⦇carrier := S⦈) a) = X_plus a"
proof-
have 20: "(X_poly_plus (R⦇carrier := S⦈) a) = (λk. if k = (0::nat) then a else
(if k = 1 then 𝟭 else 𝟬))"
using a_closed assms UP_cring.cfs_X_plus[of "R⦇carrier := S⦈" a] R.carrier_update
UP_ring_subring(1) by auto
have 21: "X_plus a = (λk. if k = (0::nat) then a else
(if k = 1 then 𝟭 else 𝟬))"
using cfs_X_plus[of a] a_closed
by blast
show ?thesis apply(rule ext) using 20 21
by auto
qed
show ?thesis
unfolding taylor_expansion_def using 0 1 2 assms UP_subring_compose[of S g "X_plus a"]
by (simp add: UP_subring_compose)
qed
lemma(in UP_cring) UP_subring_taylor_closed:
assumes "subring S R"
assumes "g ∈ carrier (UP R)"
assumes "⋀n. g n ∈ S"
assumes "a ∈ S"
shows "taylor_expansion R a g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
proof-
have "g ∈ carrier (UP (R⦇carrier := S⦈))"
by (metis P_def R.carrier_update(1) R.carrier_update(2) UP_cring.UP_car_memI' UP_ring_subring(1) assms(1) assms(2) assms(3) deg_leE)
then show ?thesis
using assms UP_cring.taylor_def[of "R⦇carrier := S⦈"] UP_subring_taylor[of S g a]
UP_cring.taylor_closed[of "R ⦇ carrier := S ⦈" g a] UP_ring_subring(1)[of S] by simp
qed
lemma(in UP_cring) UP_subring_taylor_closed':
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "a ∈ S"
shows "taylor_expansion R a g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
using UP_subring_taylor_closed assms UP_cring.UP_car_memE[of "R ⦇ carrier := S ⦈" g] R.carrier_update[of S]
UP_ring_subring(1) UP_ring_subring_car_subset' by auto
lemma(in UP_cring) UP_subring_taylor':
assumes "subring S R"
assumes "g ∈ carrier (UP R)"
assumes "⋀n. g n ∈ S"
assumes "a ∈ S"
shows "taylor_expansion R a g n ∈ S"
using assms UP_subring_taylor R.carrier_update[of S] UP_cring.taylor_closed[of "R ⦇ carrier := S ⦈"]
using UP_cring.taylor_expansion_cf_closed UP_ring_subring(1) poly_cfs_subring by metis
lemma(in UP_cring) UP_subring_deriv:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "a ∈ S"
shows "deriv g a= UP_cring.deriv (R ⦇ carrier := S ⦈) g a"
proof-
have 0: "(⋀n. g n ∈ S)"
using assms UP_ring_subring_car by blast
thus ?thesis
unfolding derivative_def using 0 UP_ring_subring_car_subset[of S] assms UP_subring_taylor[of S g a]
by (simp add: subset_iff)
qed
lemma(in UP_cring) UP_subring_deriv_closed:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "a ∈ S"
shows "deriv g a ∈ S"
using assms UP_cring.deriv_closed[of "R ⦇ carrier := S ⦈" g a] UP_subring_deriv[of S g a]
UP_ring_subring_car_subset[of S] UP_ring_subring[of S]
by simp
lemma(in UP_cring) poly_shift_subring_closed:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
shows "poly_shift g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
using UP_cring.poly_shift_closed[of "R ⦇ carrier := S ⦈" g] assms UP_ring_subring[of S]
by simp
lemma(in UP_cring) UP_subring_taylor_appr:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "a ∈ S"
assumes "b ∈ S"
shows "∃c ∈ S. to_fun g a= to_fun g b ⊕ (deriv g b)⊗ (a ⊖ b) ⊕ (c ⊗ (a ⊖ b)[^](2::nat))"
proof-
have a_closed: "a ∈ carrier R"
using assms subringE by blast
have b_closed: "b ∈ carrier R"
using assms subringE by blast
have g_closed: " g ∈ carrier (UP R)"
using UP_ring_subring_car_subset[of S] assms by blast
have 0: "to_fun (shift 2 (T⇘b⇙ g)) (a ⊖ b) = to_fun (shift 2 (T⇘b⇙ g)) (a ⊖ b)"
by simp
have 1: "to_fun g b = to_fun g b"
by simp
have 2: "deriv g b = deriv g b"
by simp
have 3: "to_fun g a = to_fun g b ⊕ deriv g b ⊗ (a ⊖ b) ⊕ to_fun (shift 2 (T⇘b⇙ g)) (a ⊖ b) ⊗ (a ⊖ b) [^] (2::nat)"
using taylor_deg_1_expansion[of g b a "to_fun (shift 2 (T⇘b⇙ g)) (a ⊖ b)" "to_fun g b" "deriv g b"]
assms a_closed b_closed g_closed 0 1 2 unfolding P_def by blast
have 4: "to_fun (shift 2 (T⇘b⇙ g)) (a ⊖ b) ∈ S"
proof-
have 0: "(2::nat) = Suc (Suc 0)"
by simp
have 1: "a ⊖ b ∈ S"
using assms unfolding a_minus_def
by (simp add: subringE(5) subringE(7))
have 2: "poly_shift (T⇘b⇙ g) ∈ carrier (UP (R⦇carrier := S⦈))"
using poly_shift_subring_closed[of S "taylor_expansion R b g"] UP_ring_subring[of S]
UP_subring_taylor_closed'[of S g b] assms unfolding taylor_def
by blast
hence 3: "poly_shift (poly_shift (T⇘b⇙ g)) ∈ carrier (UP (R⦇carrier := S⦈))"
using UP_cring.poly_shift_closed[of "R⦇carrier := S⦈" "(poly_shift (T⇘b⇙ g))"]
unfolding taylor_def
using assms(1) poly_shift_subring_closed by blast
have 4: "to_fun (poly_shift (poly_shift (T⇘b⇙ g))) (a ⊖ b) ∈ S"
using 1 2 3 0 UP_subring_eval_closed[of S "poly_shift (poly_shift (T⇘b⇙ g))" "a ⊖ b"]
UP_cring.poly_shift_closed[of "R⦇carrier := S⦈"] assms
by blast
then show ?thesis
by (simp add: numeral_2_eq_2)
qed
obtain c where c_def: "c = to_fun (shift 2 (T⇘b⇙ g)) (a ⊖ b)"
by blast
have 5: "c ∈ S ∧ to_fun g a = to_fun g b ⊕ deriv g b ⊗ (a ⊖ b) ⊕ c ⊗ (a ⊖ b) [^] (2::nat)"
unfolding c_def using 3 4 by blast
thus ?thesis using c_def 4 by blast
qed
lemma(in UP_cring) UP_subring_taylor_appr':
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
assumes "a ∈ S"
assumes "b ∈ S"
shows "∃c c' c''. c ∈ S ∧ c' ∈ S ∧ c'' ∈ S ∧ to_fun g a= c ⊕ c'⊗ (a ⊖ b) ⊕ (c'' ⊗ (a ⊖ b)[^](2::nat))"
using UP_subring_taylor_appr[of S g a b] assms UP_subring_deriv_closed[of S g b] UP_subring_eval_closed[of S g b]
by blast
lemma (in UP_cring) pderiv_cfs:
assumes"g ∈ carrier (UP R)"
shows "pderiv g n = [Suc n]⋅(g (Suc n))"
unfolding pderiv_def
using n_mult_closed[of g] assms poly_shift_cfs[of "n_mult g" n]
unfolding P_def n_mult_def by blast
lemma(in ring) subring_add_pow:
assumes "subring S R"
assumes "a ∈ S"
shows "[(n::nat)] ⋅⇘R⦇carrier := S⦈⇙ a = [(n::nat)] ⋅a"
proof-
have 0: "a ∈ carrier R"
using assms(1) assms(2) subringE(1) by blast
have 1: "a ∈ carrier (R⦇carrier := S⦈)"
by (simp add: assms(2))
show ?thesis
apply(induction n)
using assms 0 1 carrier_update[of S]
apply (simp add: add_pow_def)
using assms 0 1 carrier_update[of S]
by (simp add: add_pow_def)
qed
lemma(in UP_cring) UP_subring_pderiv_equal:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
shows "pderiv g = UP_cring.pderiv (R⦇carrier := S⦈) g"
proof fix n
show "pderiv g n = UP_cring.pderiv (R⦇carrier := S⦈) g n"
using UP_cring.pderiv_cfs[of "R ⦇ carrier := S ⦈" g n] pderiv_cfs[of g n]
assms R.subring_add_pow[of S "g (Suc n)" "Suc n"]
by (simp add: UP_ring_subring(1) UP_ring_subring_car)
qed
lemma(in UP_cring) UP_subring_pderiv_closed:
assumes "subring S R"
assumes "g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
shows "pderiv g ∈ carrier (UP (R ⦇ carrier := S ⦈))"
using assms UP_cring.pderiv_closed[of "R ⦇ carrier := S ⦈" g] R.carrier_update(1) UP_ring_subring(1)
UP_subring_pderiv_equal by auto
lemma(in UP_cring) UP_subring_pderiv_closed':
assumes "subring S R"
assumes "g ∈ carrier (UP R)"
assumes "⋀n. g n ∈ S"
shows "⋀n. pderiv g n ∈ S"
using assms UP_subring_pderiv_closed[of S g] poly_cfs_subring[of S g] UP_ring_subring_car
by blast
lemma(in UP_cring) taylor_deg_one_expansion_subring:
assumes "f ∈ carrier (UP R)"
assumes "subring S R"
assumes "⋀i. f i ∈ S"
assumes "a ∈ S"
assumes "b ∈ S"
shows "∃c ∈ S. to_fun f b = (to_fun f a) ⊕ (deriv f a) ⊗ (b ⊖ a) ⊕ (c ⊗ (b ⊖ a)[^](2::nat))"
apply(rule UP_subring_taylor_appr, rule assms)
using assms poly_cfs_subring apply blast
by(rule assms, rule assms)
lemma(in UP_cring) taylor_deg_one_expansion_subring':
assumes "f ∈ carrier (UP R)"
assumes "subring S R"
assumes "⋀i. f i ∈ S"
assumes "a ∈ S"
assumes "b ∈ S"
shows "∃c ∈ S. to_fun f b = (to_fun f a) ⊕ (to_fun (pderiv f) a) ⊗ (b ⊖ a) ⊕ (c ⊗ (b ⊖ a)[^](2::nat))"
proof-
have "S ⊆ carrier R"
using assms subringE(1) by blast
hence 0: "deriv f a = to_fun (pderiv f) a"
using assms pderiv_eval_deriv[of f a] unfolding P_def by blast
show ?thesis
using assms taylor_deg_one_expansion_subring[of f S a b]
unfolding 0 by blast
qed
end