more standard name bindings (amending 5bf71b4da706): avoid odd full_name like "Orderings.class.Orderings.preorder.of_class.intro" with many redundant name space accesses;
(* Author: Andreas Lochbihler, ETH Zurich Author: Florian Haftmann, TU Muenchen with some material ported from HOL Light by LCP*)section \<open>Polynomial mapping: combination of almost everywhere zero functions with an algebraic view\<close>theory Poly_Mappingimports Groups_Big_Fun Fun_Lexorder More_Listbeginsubsection \<open>Preliminary: auxiliary operations for \emph{almost everywhere zero}\<close>text \<open> A central notion for polynomials are functions being \emph{almost everywhere zero}. For these we provide some auxiliary definitions and lemmas.\<close>lemma finite_mult_not_eq_zero_leftI: fixes f :: "'b \<Rightarrow> 'a :: mult_zero" assumes "finite {a. f a \<noteq> 0}" shows "finite {a. g a * f a \<noteq> 0}"proof - have "{a. g a * f a \<noteq> 0} \<subseteq> {a. f a \<noteq> 0}" by auto then show ?thesis using assms by (rule finite_subset)qedlemma finite_mult_not_eq_zero_rightI: fixes f :: "'b \<Rightarrow> 'a :: mult_zero" assumes "finite {a. f a \<noteq> 0}" shows "finite {a. f a * g a \<noteq> 0}"proof - have "{a. f a * g a \<noteq> 0} \<subseteq> {a. f a \<noteq> 0}" by auto then show ?thesis using assms by (rule finite_subset)qedlemma finite_mult_not_eq_zero_prodI: fixes f g :: "'a \<Rightarrow> 'b::semiring_0" assumes "finite {a. f a \<noteq> 0}" (is "finite ?F") assumes "finite {b. g b \<noteq> 0}" (is "finite ?G") shows "finite {(a, b). f a * g b \<noteq> 0}"proof - from assms have "finite (?F \<times> ?G)" by blast then have "finite {(a, b). f a \<noteq> 0 \<and> g b \<noteq> 0}" by simp then show ?thesis by (rule rev_finite_subset) autoqedlemma finite_not_eq_zero_sumI: fixes f g :: "'a::monoid_add \<Rightarrow> 'b::semiring_0" assumes "finite {a. f a \<noteq> 0}" (is "finite ?F") assumes "finite {b. g b \<noteq> 0}" (is "finite ?G") shows "finite {a + b | a b. f a \<noteq> 0 \<and> g b \<noteq> 0}" (is "finite ?FG")proof - from assms have "finite (?F \<times> ?G)" by (simp add: finite_cartesian_product_iff) then have "finite (case_prod plus ` (?F \<times> ?G))" by (rule finite_imageI) also have "case_prod plus ` (?F \<times> ?G) = ?FG" by auto finally show ?thesis by simpqedlemma finite_mult_not_eq_zero_sumI: fixes f g :: "'a::monoid_add \<Rightarrow> 'b::semiring_0" assumes "finite {a. f a \<noteq> 0}" assumes "finite {b. g b \<noteq> 0}" shows "finite {a + b | a b. f a * g b \<noteq> 0}"proof - from assms have "finite {a + b | a b. f a \<noteq> 0 \<and> g b \<noteq> 0}" by (rule finite_not_eq_zero_sumI) then show ?thesis by (rule rev_finite_subset) (auto dest: mult_not_zero)qedlemma finite_Sum_any_not_eq_zero_weakenI: assumes "finite {a. \<exists>b. f a b \<noteq> 0}" shows "finite {a. Sum_any (f a) \<noteq> 0}"proof - have "{a. Sum_any (f a) \<noteq> 0} \<subseteq> {a. \<exists>b. f a b \<noteq> 0}" by (auto elim: Sum_any.not_neutral_obtains_not_neutral) then show ?thesis using assms by (rule finite_subset)qedcontext zerobegindefinition "when" :: "'a \<Rightarrow> bool \<Rightarrow> 'a" (infixl "when" 20)where "(a when P) = (if P then a else 0)"text \<open> Case distinctions always complicate matters, particularly when nested. The @{const "when"} operation allows to minimise these if @{term 0} is the false-case value and makes proof obligations much more readable.\<close>lemma "when" [simp]: "P \<Longrightarrow> (a when P) = a" "\<not> P \<Longrightarrow> (a when P) = 0" by (simp_all add: when_def)lemma when_simps [simp]: "(a when True) = a" "(a when False) = 0" by simp_alllemma when_cong: assumes "P \<longleftrightarrow> Q" and "Q \<Longrightarrow> a = b" shows "(a when P) = (b when Q)" using assms by (simp add: when_def)lemma zero_when [simp]: "(0 when P) = 0" by (simp add: when_def)lemma when_when: "(a when P when Q) = (a when P \<and> Q)" by (cases Q) simp_alllemma when_commute: "(a when Q when P) = (a when P when Q)" by (simp add: when_when conj_commute)lemma when_neq_zero [simp]: "(a when P) \<noteq> 0 \<longleftrightarrow> P \<and> a \<noteq> 0" by (cases P) simp_allendcontext monoid_addbeginlemma when_add_distrib: "(a + b when P) = (a when P) + (b when P)" by (simp add: when_def)endcontext semiring_1beginlemma zero_power_eq: "0 ^ n = (1 when n = 0)" by (simp add: power_0_left)endcontext comm_monoid_addbeginlemma Sum_any_when_equal [simp]: "(\<Sum>a. (f a when a = b)) = f b" by (simp add: when_def)lemma Sum_any_when_equal' [simp]: "(\<Sum>a. (f a when b = a)) = f b" by (simp add: when_def)lemma Sum_any_when_independent: "(\<Sum>a. g a when P) = ((\<Sum>a. g a) when P)" by (cases P) simp_alllemma Sum_any_when_dependent_prod_right: "(\<Sum>(a, b). g a when b = h a) = (\<Sum>a. g a)"proof - have "inj_on (\<lambda>a. (a, h a)) {a. g a \<noteq> 0}" by (rule inj_onI) auto then show ?thesis unfolding Sum_any.expand_set by (rule sum.reindex_cong) autoqedlemma Sum_any_when_dependent_prod_left: "(\<Sum>(a, b). g b when a = h b) = (\<Sum>b. g b)"proof - have "(\<Sum>(a, b). g b when a = h b) = (\<Sum>(b, a). g b when a = h b)" by (rule Sum_any.reindex_cong [of prod.swap]) (simp_all add: fun_eq_iff) then show ?thesis by (simp add: Sum_any_when_dependent_prod_right)qedendcontext cancel_comm_monoid_addbeginlemma when_diff_distrib: "(a - b when P) = (a when P) - (b when P)" by (simp add: when_def)endcontext group_addbeginlemma when_uminus_distrib: "(- a when P) = - (a when P)" by (simp add: when_def)endcontext mult_zerobeginlemma mult_when: "a * (b when P) = (a * b when P)" by (cases P) simp_alllemma when_mult: "(a when P) * b = (a * b when P)" by (cases P) simp_allendsubsection \<open>Type definition\<close>text \<open> The following type is of central importance:\<close>typedef (overloaded) ('a, 'b) poly_mapping ("(_ \<Rightarrow>\<^sub>0 /_)" [1, 0] 0) = "{f :: 'a \<Rightarrow> 'b::zero. finite {x. f x \<noteq> 0}}" morphisms lookup Abs_poly_mappingproof - have "(\<lambda>_::'a. (0 :: 'b)) \<in> ?poly_mapping" by simp then show ?thesis by (blast intro!: exI)qeddeclare lookup_inverse [simp]declare lookup_inject [simp]lemma lookup_Abs_poly_mapping [simp]: "finite {x. f x \<noteq> 0} \<Longrightarrow> lookup (Abs_poly_mapping f) = f" using Abs_poly_mapping_inverse [of f] by simplemma finite_lookup [simp]: "finite {k. lookup f k \<noteq> 0}" using poly_mapping.lookup [of f] by simplemma finite_lookup_nat [simp]: fixes f :: "'a \<Rightarrow>\<^sub>0 nat" shows "finite {k. 0 < lookup f k}" using poly_mapping.lookup [of f] by simplemma poly_mapping_eqI: assumes "\<And>k. lookup f k = lookup g k" shows "f = g" using assms unfolding poly_mapping.lookup_inject [symmetric] by blastlemma poly_mapping_eq_iff: "a = b \<longleftrightarrow> lookup a = lookup b" by autotext \<open> We model the universe of functions being \emph{almost everywhere zero} by means of a separate type @{typ "('a, 'b) poly_mapping"}. For convenience we provide a suggestive infix syntax which is a variant of the usual function space syntax. Conversion between both types happens through the morphisms \begin{quote} @{term_type lookup} \end{quote} \begin{quote} @{term_type Abs_poly_mapping} \end{quote} satisfying \begin{quote} @{thm lookup_inverse} \end{quote} \begin{quote} @{thm lookup_Abs_poly_mapping} \end{quote} Luckily, we have rarely to deal with those low-level morphisms explicitly but rely on Isabelle's \emph{lifting} package with its method \<open>transfer\<close> and its specification tool \<open>lift_definition\<close>.\<close>setup_lifting type_definition_poly_mappingcode_datatype Abs_poly_mapping\<comment>\<open>FIXME? workaround for preventing \<open>code_abstype\<close> setup\<close>text \<open> @{typ "'a \<Rightarrow>\<^sub>0 'b"} serves distinctive purposes: \begin{enumerate} \item A clever nesting as @{typ "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"} later in theory \<open>MPoly\<close> gives a suitable representation type for polynomials \emph{almost for free}: Interpreting @{typ "nat \<Rightarrow>\<^sub>0 nat"} as a mapping from variable identifiers to exponents yields monomials, and the whole type maps monomials to coefficients. Lets call this the \emph{ultimate interpretation}. \item A further more specialised type isomorphic to @{typ "nat \<Rightarrow>\<^sub>0 'a"} is apt to direct implementation using code generation \<^cite>\<open>"Haftmann-Nipkow:2010:code"\<close>. \end{enumerate} Note that despite the names \emph{mapping} and \emph{lookup} suggest something implementation-near, it is best to keep @{typ "'a \<Rightarrow>\<^sub>0 'b"} as an abstract \emph{algebraic} type providing operations like \emph{addition}, \emph{multiplication} without any notion of key-order, data structures etc. This implementations-specific notions are easily introduced later for particular implementations but do not provide any gain for specifying logical properties of polynomials.\<close>subsection \<open>Additive structure\<close>text \<open> The additive structure covers the usual operations \<open>0\<close>, \<open>+\<close> and (unary and binary) \<open>-\<close>. Recalling the ultimate interpretation, it is obvious that these have just lift the corresponding operations on values to mappings. Isabelle has a rich hierarchy of algebraic type classes, and in such situations of pointwise lifting a typical pattern is to have instantiations for a considerable number of type classes. The operations themselves are specified using \<open>lift_definition\<close>, where the proofs of the \emph{almost everywhere zero} property can be significantly involved. The @{const lookup} operation is supposed to be usable explicitly (unless in other situations where the morphisms between types are somehow internal to the \emph{lifting} package). Hence it is good style to provide explicit rewrite rules how @{const lookup} acts on operations immediately.\<close>instantiation poly_mapping :: (type, zero) zerobeginlift_definition zero_poly_mapping :: "'a \<Rightarrow>\<^sub>0 'b" is "\<lambda>k. 0" by simpinstance ..endlemma Abs_poly_mapping [simp]: "Abs_poly_mapping (\<lambda>k. 0) = 0" by (simp add: zero_poly_mapping.abs_eq)lemma lookup_zero [simp]: "lookup 0 k = 0" by transfer ruleinstantiation poly_mapping :: (type, monoid_add) monoid_addbeginlift_definition plus_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b" is "\<lambda>f1 f2 k. f1 k + f2 k"proof - fix f1 f2 :: "'a \<Rightarrow> 'b" assume "finite {k. f1 k \<noteq> 0}" and "finite {k. f2 k \<noteq> 0}" then have "finite ({k. f1 k \<noteq> 0} \<union> {k. f2 k \<noteq> 0})" by auto moreover have "{x. f1 x + f2 x \<noteq> 0} \<subseteq> {k. f1 k \<noteq> 0} \<union> {k. f2 k \<noteq> 0}" by auto ultimately show "finite {x. f1 x + f2 x \<noteq> 0}" by (blast intro: finite_subset)qedinstance by intro_classes (transfer, simp add: fun_eq_iff ac_simps)+endlemma lookup_add: "lookup (f + g) k = lookup f k + lookup g k" by transfer ruleinstance poly_mapping :: (type, comm_monoid_add) comm_monoid_add by intro_classes (transfer, simp add: fun_eq_iff ac_simps)+lemma lookup_sum: "lookup (sum pp X) i = sum (\<lambda>x. lookup (pp x) i) X" by (induction rule: infinite_finite_induct) (auto simp: lookup_add)(*instance poly_mapping :: (type, "{monoid_add, cancel_semigroup_add}") cancel_semigroup_add by intro_classes (transfer, simp add: fun_eq_iff)+*)instantiation poly_mapping :: (type, cancel_comm_monoid_add) cancel_comm_monoid_addbeginlift_definition minus_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b" is "\<lambda>f1 f2 k. f1 k - f2 k"proof - fix f1 f2 :: "'a \<Rightarrow> 'b" assume "finite {k. f1 k \<noteq> 0}" and "finite {k. f2 k \<noteq> 0}" then have "finite ({k. f1 k \<noteq> 0} \<union> {k. f2 k \<noteq> 0})" by auto moreover have "{x. f1 x - f2 x \<noteq> 0} \<subseteq> {k. f1 k \<noteq> 0} \<union> {k. f2 k \<noteq> 0}" by auto ultimately show "finite {x. f1 x - f2 x \<noteq> 0}" by (blast intro: finite_subset)qedinstance by intro_classes (transfer, simp add: fun_eq_iff diff_diff_add)+endinstantiation poly_mapping :: (type, ab_group_add) ab_group_addbeginlift_definition uminus_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b" is uminus by simpinstance by intro_classes (transfer, simp add: fun_eq_iff ac_simps)+endlemma lookup_uminus [simp]: "lookup (- f) k = - lookup f k" by transfer simplemma lookup_minus: "lookup (f - g) k = lookup f k - lookup g k" by transfer rulesubsection \<open>Multiplicative structure\<close>instantiation poly_mapping :: (zero, zero_neq_one) zero_neq_onebeginlift_definition one_poly_mapping :: "'a \<Rightarrow>\<^sub>0 'b" is "\<lambda>k. 1 when k = 0" by simpinstance by intro_classes (transfer, simp add: fun_eq_iff)endlemma lookup_one: "lookup 1 k = (1 when k = 0)" by transfer rulelemma lookup_one_zero [simp]: "lookup 1 0 = 1" by transfer simpdefinition prod_fun :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a::monoid_add \<Rightarrow> 'b::semiring_0"where "prod_fun f1 f2 k = (\<Sum>l. f1 l * (\<Sum>q. (f2 q when k = l + q)))"lemma prod_fun_unfold_prod: fixes f g :: "'a :: monoid_add \<Rightarrow> 'b::semiring_0" assumes fin_f: "finite {a. f a \<noteq> 0}" assumes fin_g: "finite {b. g b \<noteq> 0}" shows "prod_fun f g k = (\<Sum>(a, b). f a * g b when k = a + b)"proof - let ?C = "{a. f a \<noteq> 0} \<times> {b. g b \<noteq> 0}" from fin_f fin_g have "finite ?C" by blast moreover have "{a. \<exists>b. (f a * g b when k = a + b) \<noteq> 0} \<times> {b. \<exists>a. (f a * g b when k = a + b) \<noteq> 0} \<subseteq> {a. f a \<noteq> 0} \<times> {b. g b \<noteq> 0}" by auto ultimately show ?thesis using fin_g by (auto simp add: prod_fun_def Sum_any.cartesian_product [of "{a. f a \<noteq> 0} \<times> {b. g b \<noteq> 0}"] Sum_any_right_distrib mult_when)qedlemma finite_prod_fun: fixes f1 f2 :: "'a :: monoid_add \<Rightarrow> 'b :: semiring_0" assumes fin1: "finite {l. f1 l \<noteq> 0}" and fin2: "finite {q. f2 q \<noteq> 0}" shows "finite {k. prod_fun f1 f2 k \<noteq> 0}"proof - have *: "finite {k. (\<exists>l. f1 l \<noteq> 0 \<and> (\<exists>q. f2 q \<noteq> 0 \<and> k = l + q))}" using assms by simp { fix k l have "{q. (f2 q when k = l + q) \<noteq> 0} \<subseteq> {q. f2 q \<noteq> 0 \<and> k = l + q}" by auto with fin2 have "sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q} = (\<Sum>q. (f2 q when k = l + q))" by (simp add: Sum_any.expand_superset [of "{q. f2 q \<noteq> 0 \<and> k = l + q}"]) } note aux = this have "{k. (\<Sum>l. f1 l * sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q}) \<noteq> 0} \<subseteq> {k. (\<exists>l. f1 l * sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q} \<noteq> 0)}" by (auto elim!: Sum_any.not_neutral_obtains_not_neutral) also have "\<dots> \<subseteq> {k. (\<exists>l. f1 l \<noteq> 0 \<and> sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q} \<noteq> 0)}" by (auto dest: mult_not_zero) also have "\<dots> \<subseteq> {k. (\<exists>l. f1 l \<noteq> 0 \<and> (\<exists>q. f2 q \<noteq> 0 \<and> k = l + q))}" by (auto elim!: sum.not_neutral_contains_not_neutral) finally have "finite {k. (\<Sum>l. f1 l * sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q}) \<noteq> 0}" using * by (rule finite_subset) with aux have "finite {k. (\<Sum>l. f1 l * (\<Sum>q. (f2 q when k = l + q))) \<noteq> 0}" by simp with fin2 show ?thesis by (simp add: prod_fun_def)qedinstantiation poly_mapping :: (monoid_add, semiring_0) semiring_0beginlift_definition times_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b" is prod_funby(rule finite_prod_fun)instanceproof fix a b c :: "'a \<Rightarrow>\<^sub>0 'b" show "a * b * c = a * (b * c)" proof transfer fix f g h :: "'a \<Rightarrow> 'b" assume fin_f: "finite {a. f a \<noteq> 0}" (is "finite ?F") assume fin_g: "finite {b. g b \<noteq> 0}" (is "finite ?G") assume fin_h: "finite {c. h c \<noteq> 0}" (is "finite ?H") from fin_f fin_g have fin_fg: "finite {(a, b). f a * g b \<noteq> 0}" (is "finite ?FG") by (rule finite_mult_not_eq_zero_prodI) from fin_g fin_h have fin_gh: "finite {(b, c). g b * h c \<noteq> 0}" (is "finite ?GH") by (rule finite_mult_not_eq_zero_prodI) from fin_f fin_g have fin_fg': "finite {a + b | a b. f a * g b \<noteq> 0}" (is "finite ?FG'") by (rule finite_mult_not_eq_zero_sumI) then have fin_fg'': "finite {d. (\<Sum>(a, b). f a * g b when d = a + b) \<noteq> 0}" by (auto intro: finite_Sum_any_not_eq_zero_weakenI) from fin_g fin_h have fin_gh': "finite {b + c | b c. g b * h c \<noteq> 0}" (is "finite ?GH'") by (rule finite_mult_not_eq_zero_sumI) then have fin_gh'': "finite {d. (\<Sum>(b, c). g b * h c when d = b + c) \<noteq> 0}" by (auto intro: finite_Sum_any_not_eq_zero_weakenI) show "prod_fun (prod_fun f g) h = prod_fun f (prod_fun g h)" (is "?lhs = ?rhs") proof fix k from fin_f fin_g fin_h fin_fg'' have "?lhs k = (\<Sum>(ab, c). (\<Sum>(a, b). f a * g b when ab = a + b) * h c when k = ab + c)" by (simp add: prod_fun_unfold_prod) also have "\<dots> = (\<Sum>(ab, c). (\<Sum>(a, b). f a * g b * h c when k = ab + c when ab = a + b))" apply (subst Sum_any_left_distrib) using fin_fg apply (simp add: split_def) apply (subst Sum_any_when_independent [symmetric]) apply (simp add: when_when when_mult mult_when split_def conj_commute) done also have "\<dots> = (\<Sum>(ab, c, a, b). f a * g b * h c when k = ab + c when ab = a + b)" apply (subst Sum_any.cartesian_product2 [of "(?FG' \<times> ?H) \<times> ?FG"]) apply (auto simp add: finite_cartesian_product_iff fin_fg fin_fg' fin_h dest: mult_not_zero) done also have "\<dots> = (\<Sum>(ab, c, a, b). f a * g b * h c when k = a + b + c when ab = a + b)" by (rule Sum_any.cong) (simp add: split_def when_def) also have "\<dots> = (\<Sum>(ab, cab). (case cab of (c, a, b) \<Rightarrow> f a * g b * h c when k = a + b + c) when ab = (case cab of (c, a, b) \<Rightarrow> a + b))" by (simp add: split_def) also have "\<dots> = (\<Sum>(c, a, b). f a * g b * h c when k = a + b + c)" by (simp add: Sum_any_when_dependent_prod_left) also have "\<dots> = (\<Sum>(bc, cab). (case cab of (c, a, b) \<Rightarrow> f a * g b * h c when k = a + b + c) when bc = (case cab of (c, a, b) \<Rightarrow> b + c))" by (simp add: Sum_any_when_dependent_prod_left) also have "\<dots> = (\<Sum>(bc, c, a, b). f a * g b * h c when k = a + b + c when bc = b + c)" by (simp add: split_def) also have "\<dots> = (\<Sum>(bc, c, a, b). f a * g b * h c when bc = b + c when k = a + bc)" by (rule Sum_any.cong) (simp add: split_def when_def ac_simps) also have "\<dots> = (\<Sum>(a, bc, b, c). f a * g b * h c when bc = b + c when k = a + bc)" proof - have "bij (\<lambda>(a, d, b, c). (d, c, a, b))" by (auto intro!: bijI injI surjI [of _ "\<lambda>(d, c, a, b). (a, d, b, c)"] simp add: split_def) then show ?thesis by (rule Sum_any.reindex_cong) auto qed also have "\<dots> = (\<Sum>(a, bc). (\<Sum>(b, c). f a * g b * h c when bc = b + c when k = a + bc))" apply (subst Sum_any.cartesian_product2 [of "(?F \<times> ?GH') \<times> ?GH"]) apply (auto simp add: finite_cartesian_product_iff fin_f fin_gh fin_gh' ac_simps dest: mult_not_zero) done also have "\<dots> = (\<Sum>(a, bc). f a * (\<Sum>(b, c). g b * h c when bc = b + c) when k = a + bc)" apply (subst Sum_any_right_distrib) using fin_gh apply (simp add: split_def) apply (subst Sum_any_when_independent [symmetric]) apply (simp add: when_when when_mult mult_when split_def ac_simps) done also from fin_f fin_g fin_h fin_gh'' have "\<dots> = ?rhs k" by (simp add: prod_fun_unfold_prod) finally show "?lhs k = ?rhs k" . qed qed show "(a + b) * c = a * c + b * c" proof transfer fix f g h :: "'a \<Rightarrow> 'b" assume fin_f: "finite {k. f k \<noteq> 0}" assume fin_g: "finite {k. g k \<noteq> 0}" assume fin_h: "finite {k. h k \<noteq> 0}" show "prod_fun (\<lambda>k. f k + g k) h = (\<lambda>k. prod_fun f h k + prod_fun g h k)" apply (rule ext) apply (auto simp add: prod_fun_def algebra_simps) apply (subst Sum_any.distrib) using fin_f fin_g apply (auto intro: finite_mult_not_eq_zero_rightI) done qed show "a * (b + c) = a * b + a * c" proof transfer fix f g h :: "'a \<Rightarrow> 'b" assume fin_f: "finite {k. f k \<noteq> 0}" assume fin_g: "finite {k. g k \<noteq> 0}" assume fin_h: "finite {k. h k \<noteq> 0}" show "prod_fun f (\<lambda>k. g k + h k) = (\<lambda>k. prod_fun f g k + prod_fun f h k)" apply (rule ext) apply (auto simp add: prod_fun_def Sum_any.distrib algebra_simps when_add_distrib) apply (subst Sum_any.distrib) apply (simp_all add: algebra_simps) apply (auto intro: fin_g fin_h) apply (subst Sum_any.distrib) apply (simp_all add: algebra_simps) using fin_f apply (rule finite_mult_not_eq_zero_rightI) using fin_f apply (rule finite_mult_not_eq_zero_rightI) done qed show "0 * a = 0" by transfer (simp add: prod_fun_def [abs_def]) show "a * 0 = 0" by transfer (simp add: prod_fun_def [abs_def])qedendlemma lookup_mult: "lookup (f * g) k = (\<Sum>l. lookup f l * (\<Sum>q. lookup g q when k = l + q))" by transfer (simp add: prod_fun_def)instance poly_mapping :: (comm_monoid_add, comm_semiring_0) comm_semiring_0proof fix a b c :: "'a \<Rightarrow>\<^sub>0 'b" show "a * b = b * a" proof transfer fix f g :: "'a \<Rightarrow> 'b" assume fin_f: "finite {a. f a \<noteq> 0}" assume fin_g: "finite {b. g b \<noteq> 0}" show "prod_fun f g = prod_fun g f" proof fix k have fin1: "\<And>l. finite {a. (f a when k = l + a) \<noteq> 0}" using fin_f by auto have fin2: "\<And>l. finite {b. (g b when k = l + b) \<noteq> 0}" using fin_g by auto from fin_f fin_g have "finite {(a, b). f a \<noteq> 0 \<and> g b \<noteq> 0}" (is "finite ?AB") by simp show "prod_fun f g k = prod_fun g f k" apply (simp add: prod_fun_def) apply (subst Sum_any_right_distrib) apply (rule fin2) apply (subst Sum_any_right_distrib) apply (rule fin1) apply (subst Sum_any.swap [of ?AB]) apply (fact \<open>finite ?AB\<close>) apply (auto simp add: mult_when ac_simps) done qed qed show "(a + b) * c = a * c + b * c" proof transfer fix f g h :: "'a \<Rightarrow> 'b" assume fin_f: "finite {k. f k \<noteq> 0}" assume fin_g: "finite {k. g k \<noteq> 0}" assume fin_h: "finite {k. h k \<noteq> 0}" show "prod_fun (\<lambda>k. f k + g k) h = (\<lambda>k. prod_fun f h k + prod_fun g h k)" apply (auto simp add: prod_fun_def fun_eq_iff algebra_simps) apply (subst Sum_any.distrib) using fin_f apply (rule finite_mult_not_eq_zero_rightI) using fin_g apply (rule finite_mult_not_eq_zero_rightI) apply simp_all done qedqedinstance poly_mapping :: (monoid_add, semiring_0_cancel) semiring_0_cancel ..instance poly_mapping :: (comm_monoid_add, comm_semiring_0_cancel) comm_semiring_0_cancel ..instance poly_mapping :: (monoid_add, semiring_1) semiring_1proof fix a :: "'a \<Rightarrow>\<^sub>0 'b" show "1 * a = a" by transfer (simp add: prod_fun_def [abs_def] when_mult) show "a * 1 = a" apply transfer apply (simp add: prod_fun_def [abs_def] Sum_any_right_distrib Sum_any_left_distrib mult_when) apply (subst when_commute) apply simp doneqedinstance poly_mapping :: (comm_monoid_add, comm_semiring_1) comm_semiring_1proof fix a :: "'a \<Rightarrow>\<^sub>0 'b" show "1 * a = a" by transfer (simp add: prod_fun_def [abs_def])qedinstance poly_mapping :: (monoid_add, semiring_1_cancel) semiring_1_cancel ..instance poly_mapping :: (monoid_add, ring) ring ..instance poly_mapping :: (comm_monoid_add, comm_ring) comm_ring ..instance poly_mapping :: (monoid_add, ring_1) ring_1 ..instance poly_mapping :: (comm_monoid_add, comm_ring_1) comm_ring_1 ..subsection \<open>Single-point mappings\<close>lift_definition single :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b::zero" is "\<lambda>k v k'. (v when k = k')" by simplemma inj_single [iff]: "inj (single k)"proof (rule injI, transfer) fix k :: 'b and a b :: "'a::zero" assume "(\<lambda>k'. a when k = k') = (\<lambda>k'. b when k = k')" then have "(\<lambda>k'. a when k = k') k = (\<lambda>k'. b when k = k') k" by (rule arg_cong) then show "a = b" by simpqedlemma lookup_single: "lookup (single k v) k' = (v when k = k')" by transfer rulelemma lookup_single_eq [simp]: "lookup (single k v) k = v" by transfer simplemma lookup_single_not_eq: "k \<noteq> k' \<Longrightarrow> lookup (single k v) k' = 0" by transfer simplemma single_zero [simp]: "single k 0 = 0" by transfer simplemma single_one [simp]: "single 0 1 = 1" by transfer simplemma single_add: "single k (a + b) = single k a + single k b" by transfer (simp add: fun_eq_iff when_add_distrib)lemma single_uminus: "single k (- a) = - single k a" by transfer (simp add: fun_eq_iff when_uminus_distrib)lemma single_diff: "single k (a - b) = single k a - single k b" by transfer (simp add: fun_eq_iff when_diff_distrib)lemma single_numeral [simp]: "single 0 (numeral n) = numeral n" by (induct n) (simp_all only: numeral.simps numeral_add single_zero single_one single_add)lemma lookup_numeral: "lookup (numeral n) k = (numeral n when k = 0)"proof - have "lookup (numeral n) k = lookup (single 0 (numeral n)) k" by simp then show ?thesis unfolding lookup_single by simpqedlemma single_of_nat [simp]: "single 0 (of_nat n) = of_nat n" by (induct n) (simp_all add: single_add)lemma lookup_of_nat: "lookup (of_nat n) k = (of_nat n when k = 0)"proof - have "lookup (of_nat n) k = lookup (single 0 (of_nat n)) k" by simp then show ?thesis unfolding lookup_single by simpqedlemma of_nat_single: "of_nat = single 0 \<circ> of_nat" by (simp add: fun_eq_iff)lemma mult_single: "single k a * single l b = single (k + l) (a * b)"proof transfer fix k l :: 'a and a b :: 'b show "prod_fun (\<lambda>k'. a when k = k') (\<lambda>k'. b when l = k') = (\<lambda>k'. a * b when k + l = k')" proof fix k' have "prod_fun (\<lambda>k'. a when k = k') (\<lambda>k'. b when l = k') k' = (\<Sum>n. a * b when l = n when k' = k + n)" by (simp add: prod_fun_def Sum_any_right_distrib mult_when when_mult) also have "\<dots> = (\<Sum>n. a * b when k' = k + n when l = n)" by (simp add: when_when conj_commute) also have "\<dots> = (a * b when k' = k + l)" by simp also have "\<dots> = (a * b when k + l = k')" by (simp add: when_def) finally show "prod_fun (\<lambda>k'. a when k = k') (\<lambda>k'. b when l = k') k' = (\<lambda>k'. a * b when k + l = k') k'" . qedqedinstance poly_mapping :: (monoid_add, semiring_char_0) semiring_char_0 by intro_classes (auto intro: inj_compose inj_of_nat simp add: of_nat_single)instance poly_mapping :: (monoid_add, ring_char_0) ring_char_0 ..lemma single_of_int [simp]: "single 0 (of_int k) = of_int k" by (cases k) (simp_all add: single_diff single_uminus)lemma lookup_of_int: "lookup (of_int l) k = (of_int l when k = 0)"proof - have "lookup (of_int l) k = lookup (single 0 (of_int l)) k" by simp then show ?thesis unfolding lookup_single by simpqedsubsection \<open>Integral domains\<close>instance poly_mapping :: ("{ordered_cancel_comm_monoid_add, linorder}", semiring_no_zero_divisors) semiring_no_zero_divisors text \<open>The @{class "linorder"} constraint is a pragmatic device for the proof --- maybe it can be dropped\<close>proof fix f g :: "'a \<Rightarrow>\<^sub>0 'b" assume "f \<noteq> 0" and "g \<noteq> 0" then show "f * g \<noteq> 0" proof transfer fix f g :: "'a \<Rightarrow> 'b" define F where "F = {a. f a \<noteq> 0}" moreover define G where "G = {a. g a \<noteq> 0}" ultimately have [simp]: "\<And>a. f a \<noteq> 0 \<longleftrightarrow> a \<in> F" "\<And>b. g b \<noteq> 0 \<longleftrightarrow> b \<in> G" by simp_all assume "finite {a. f a \<noteq> 0}" then have [simp]: "finite F" by simp assume "finite {a. g a \<noteq> 0}" then have [simp]: "finite G" by simp assume "f \<noteq> (\<lambda>a. 0)" then obtain a where "f a \<noteq> 0" by (auto simp add: fun_eq_iff) assume "g \<noteq> (\<lambda>b. 0)" then obtain b where "g b \<noteq> 0" by (auto simp add: fun_eq_iff) from \<open>f a \<noteq> 0\<close> and \<open>g b \<noteq> 0\<close> have "F \<noteq> {}" and "G \<noteq> {}" by auto note Max_F = \<open>finite F\<close> \<open>F \<noteq> {}\<close> note Max_G = \<open>finite G\<close> \<open>G \<noteq> {}\<close> from Max_F and Max_G have [simp]: "Max F \<in> F" "Max G \<in> G" by auto from Max_F Max_G have [dest!]: "\<And>a. a \<in> F \<Longrightarrow> a \<le> Max F" "\<And>b. b \<in> G \<Longrightarrow> b \<le> Max G" by auto define q where "q = Max F + Max G" have "(\<Sum>(a, b). f a * g b when q = a + b) = (\<Sum>(a, b). f a * g b when q = a + b when a \<in> F \<and> b \<in> G)" by (rule Sum_any.cong) (auto simp add: split_def when_def q_def intro: ccontr) also have "\<dots> = (\<Sum>(a, b). f a * g b when (Max F, Max G) = (a, b))" proof (rule Sum_any.cong) fix ab :: "'a \<times> 'a" obtain a b where [simp]: "ab = (a, b)" by (cases ab) simp_all have [dest!]: "a \<le> Max F \<Longrightarrow> Max F \<noteq> a \<Longrightarrow> a < Max F" "b \<le> Max G \<Longrightarrow> Max G \<noteq> b \<Longrightarrow> b < Max G" by auto show "(case ab of (a, b) \<Rightarrow> f a * g b when q = a + b when a \<in> F \<and> b \<in> G) = (case ab of (a, b) \<Rightarrow> f a * g b when (Max F, Max G) = (a, b))" by (auto simp add: split_def when_def q_def dest: add_strict_mono [of a "Max F" b "Max G"]) qed also have "\<dots> = (\<Sum>ab. (case ab of (a, b) \<Rightarrow> f a * g b) when (Max F, Max G) = ab)" unfolding split_def when_def by auto also have "\<dots> \<noteq> 0" by simp finally have "prod_fun f g q \<noteq> 0" by (simp add: prod_fun_unfold_prod) then show "prod_fun f g \<noteq> (\<lambda>k. 0)" by (auto simp add: fun_eq_iff) qedqedinstance poly_mapping :: ("{ordered_cancel_comm_monoid_add, linorder}", ring_no_zero_divisors) ring_no_zero_divisors ..instance poly_mapping :: ("{ordered_cancel_comm_monoid_add, linorder}", ring_1_no_zero_divisors) ring_1_no_zero_divisors ..instance poly_mapping :: ("{ordered_cancel_comm_monoid_add, linorder}", idom) idom ..subsection \<open>Mapping order\<close>instantiation poly_mapping :: (linorder, "{zero, linorder}") linorderbeginlift_definition less_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool" is less_fun .lift_definition less_eq_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool" is "\<lambda>f g. less_fun f g \<or> f = g" .instance proof (rule linorder.intro_of_class) show "class.linorder (less_eq :: (_ \<Rightarrow>\<^sub>0 _) \<Rightarrow> _) less" proof (rule linorder_strictI, rule order_strictI) fix f g h :: "'a \<Rightarrow>\<^sub>0 'b" show "f \<le> g \<longleftrightarrow> f < g \<or> f = g" by transfer (rule refl) show "\<not> f < f" by transfer (rule less_fun_irrefl) show "f < g \<or> f = g \<or> g < f" proof transfer fix f g :: "'a \<Rightarrow> 'b" assume "finite {k. f k \<noteq> 0}" and "finite {k. g k \<noteq> 0}" then have "finite ({k. f k \<noteq> 0} \<union> {k. g k \<noteq> 0})" by simp moreover have "{k. f k \<noteq> g k} \<subseteq> {k. f k \<noteq> 0} \<union> {k. g k \<noteq> 0}" by auto ultimately have "finite {k. f k \<noteq> g k}" by (rule rev_finite_subset) then show "less_fun f g \<or> f = g \<or> less_fun g f" by (rule less_fun_trichotomy) qed assume "f < g" then show "\<not> g < f" by transfer (rule less_fun_asym) note \<open>f < g\<close> moreover assume "g < h" ultimately show "f < h" by transfer (rule less_fun_trans) qedqedendinstance poly_mapping :: (linorder, "{ordered_comm_monoid_add, ordered_ab_semigroup_add_imp_le, linorder}") ordered_ab_semigroup_addproof (intro_classes, transfer) fix f g h :: "'a \<Rightarrow> 'b" assume *: "less_fun f g \<or> f = g" { assume "less_fun f g" then obtain k where "f k < g k" "(\<And>k'. k' < k \<Longrightarrow> f k' = g k')" by (blast elim!: less_funE) then have "h k + f k < h k + g k" "(\<And>k'. k' < k \<Longrightarrow> h k' + f k' = h k' + g k')" by simp_all then have "less_fun (\<lambda>k. h k + f k) (\<lambda>k. h k + g k)" by (blast intro: less_funI) } with * show "less_fun (\<lambda>k. h k + f k) (\<lambda>k. h k + g k) \<or> (\<lambda>k. h k + f k) = (\<lambda>k. h k + g k)" by (auto simp add: fun_eq_iff)qedinstance poly_mapping :: (linorder, "{ordered_comm_monoid_add, ordered_ab_semigroup_add_imp_le, cancel_comm_monoid_add, linorder}") linordered_cancel_ab_semigroup_add ..instance poly_mapping :: (linorder, "{ordered_comm_monoid_add, ordered_ab_semigroup_add_imp_le, cancel_comm_monoid_add, linorder}") ordered_comm_monoid_add ..instance poly_mapping :: (linorder, "{ordered_comm_monoid_add, ordered_ab_semigroup_add_imp_le, cancel_comm_monoid_add, linorder}") ordered_cancel_comm_monoid_add ..instance poly_mapping :: (linorder, linordered_ab_group_add) linordered_ab_group_add ..text \<open> For pragmatism we leave out the final elements in the hierarchy: @{class linordered_ring}, @{class linordered_ring_strict}, @{class linordered_idom}; remember that the order instance is a mere technical device, not a deeper algebraic property.\<close>subsection \<open>Fundamental mapping notions\<close>lift_definition keys :: "('a \<Rightarrow>\<^sub>0 'b::zero) \<Rightarrow> 'a set" is "\<lambda>f. {k. f k \<noteq> 0}" .lift_definition range :: "('a \<Rightarrow>\<^sub>0 'b::zero) \<Rightarrow> 'b set" is "\<lambda>f :: 'a \<Rightarrow> 'b. Set.range f - {0}" .lemma finite_keys [simp]: "finite (keys f)" by transferlemma not_in_keys_iff_lookup_eq_zero: "k \<notin> keys f \<longleftrightarrow> lookup f k = 0" by transfer simplemma lookup_not_eq_zero_eq_in_keys: "lookup f k \<noteq> 0 \<longleftrightarrow> k \<in> keys f" by transfer simplemma lookup_eq_zero_in_keys_contradict [dest]: "lookup f k = 0 \<Longrightarrow> \<not> k \<in> keys f" by (simp add: not_in_keys_iff_lookup_eq_zero)lemma finite_range [simp]: "finite (Poly_Mapping.range p)"proof transfer fix f :: "'b \<Rightarrow> 'a" assume *: "finite {x. f x \<noteq> 0}" have "Set.range f - {0} \<subseteq> f ` {x. f x \<noteq> 0}" by auto thus "finite (Set.range f - {0})" by(rule finite_subset)(rule finite_imageI[OF *])qedlemma in_keys_lookup_in_range [simp]: "k \<in> keys f \<Longrightarrow> lookup f k \<in> range f" by transfer simplemma in_keys_iff: "x \<in> (keys s) = (lookup s x \<noteq> 0)" by (transfer, simp)lemma keys_zero [simp]: "keys 0 = {}" by transfer simplemma range_zero [simp]: "range 0 = {}" by transfer autolemma keys_add: "keys (f + g) \<subseteq> keys f \<union> keys g" by transfer autolemma keys_one [simp]: "keys 1 = {0}" by transfer simplemma range_one [simp]: "range 1 = {1}" by transfer (auto simp add: when_def)lemma keys_single [simp]: "keys (single k v) = (if v = 0 then {} else {k})" by transfer simplemma range_single [simp]: "range (single k v) = (if v = 0 then {} else {v})" by transfer (auto simp add: when_def)lemma keys_mult: "keys (f * g) \<subseteq> {a + b | a b. a \<in> keys f \<and> b \<in> keys g}" apply transfer apply (auto simp add: prod_fun_def dest!: mult_not_zero elim!: Sum_any.not_neutral_obtains_not_neutral) apply blast donelemma setsum_keys_plus_distrib: assumes hom_0: "\<And>k. f k 0 = 0" and hom_plus: "\<And>k. k \<in> Poly_Mapping.keys p \<union> Poly_Mapping.keys q \<Longrightarrow> f k (Poly_Mapping.lookup p k + Poly_Mapping.lookup q k) = f k (Poly_Mapping.lookup p k) + f k (Poly_Mapping.lookup q k)" shows "(\<Sum>k\<in>Poly_Mapping.keys (p + q). f k (Poly_Mapping.lookup (p + q) k)) = (\<Sum>k\<in>Poly_Mapping.keys p. f k (Poly_Mapping.lookup p k)) + (\<Sum>k\<in>Poly_Mapping.keys q. f k (Poly_Mapping.lookup q k))" (is "?lhs = ?p + ?q")proof - let ?A = "Poly_Mapping.keys p \<union> Poly_Mapping.keys q" have "?lhs = (\<Sum>k\<in>?A. f k (Poly_Mapping.lookup p k + Poly_Mapping.lookup q k))" apply(rule sum.mono_neutral_cong_left) apply(simp_all add: Poly_Mapping.keys_add) apply(transfer fixing: f) apply(auto simp add: hom_0)[1] apply(transfer fixing: f) apply(auto simp add: hom_0)[1] done also have "\<dots> = (\<Sum>k\<in>?A. f k (Poly_Mapping.lookup p k) + f k (Poly_Mapping.lookup q k))" by(rule sum.cong)(simp_all add: hom_plus) also have "\<dots> = (\<Sum>k\<in>?A. f k (Poly_Mapping.lookup p k)) + (\<Sum>k\<in>?A. f k (Poly_Mapping.lookup q k))" (is "_ = ?p' + ?q'") by(simp add: sum.distrib) also have "?p' = ?p" by (simp add: hom_0 in_keys_iff sum.mono_neutral_cong_right) also have "?q' = ?q" by (simp add: hom_0 in_keys_iff sum.mono_neutral_cong_right) finally show ?thesis .qedsubsection \<open>Degree\<close>definition degree :: "(nat \<Rightarrow>\<^sub>0 'a::zero) \<Rightarrow> nat"where "degree f = Max (insert 0 (Suc ` keys f))"lemma degree_zero [simp]: "degree 0 = 0" unfolding degree_def by transfer simplemma degree_one [simp]: "degree 1 = 1" unfolding degree_def by transfer simplemma degree_single_zero [simp]: "degree (single k 0) = 0" unfolding degree_def by transfer simplemma degree_single_not_zero [simp]: "v \<noteq> 0 \<Longrightarrow> degree (single k v) = Suc k" unfolding degree_def by transfer simplemma degree_zero_iff [simp]: "degree f = 0 \<longleftrightarrow> f = 0"unfolding degree_def proof transfer fix f :: "nat \<Rightarrow> 'a" assume "finite {n. f n \<noteq> 0}" then have fin: "finite (insert 0 (Suc ` {n. f n \<noteq> 0}))" by auto show "Max (insert 0 (Suc ` {n. f n \<noteq> 0})) = 0 \<longleftrightarrow> f = (\<lambda>n. 0)" (is "?P \<longleftrightarrow> ?Q") proof assume ?P have "{n. f n \<noteq> 0} = {}" proof (rule ccontr) assume "{n. f n \<noteq> 0} \<noteq> {}" then obtain n where "n \<in> {n. f n \<noteq> 0}" by blast then have "{n. f n \<noteq> 0} = insert n {n. f n \<noteq> 0}" by auto then have "Suc ` {n. f n \<noteq> 0} = insert (Suc n) (Suc ` {n. f n \<noteq> 0})" by auto with \<open>?P\<close> have "Max (insert 0 (insert (Suc n) (Suc ` {n. f n \<noteq> 0}))) = 0" by simp then have "Max (insert (Suc n) (insert 0 (Suc ` {n. f n \<noteq> 0}))) = 0" by (simp add: insert_commute) with fin have "max (Suc n) (Max (insert 0 (Suc ` {n. f n \<noteq> 0}))) = 0" by simp then show False by simp qed then show ?Q by (simp add: fun_eq_iff) next assume ?Q then show ?P by simp qedqedlemma degree_greater_zero_in_keys: assumes "0 < degree f" shows "degree f - 1 \<in> keys f"proof - from assms have "keys f \<noteq> {}" by (auto simp add: degree_def) then show ?thesis unfolding degree_def by (simp add: mono_Max_commute [symmetric] mono_Suc)qedlemma in_keys_less_degree: "n \<in> keys f \<Longrightarrow> n < degree f"unfolding degree_def by transfer (auto simp add: Max_gr_iff)lemma beyond_degree_lookup_zero: "degree f \<le> n \<Longrightarrow> lookup f n = 0" unfolding degree_def by transfer autolemma degree_add: "degree (f + g) \<le> max (degree f) (Poly_Mapping.degree g)"unfolding degree_def proof transfer fix f g :: "nat \<Rightarrow> 'a" assume f: "finite {x. f x \<noteq> 0}" assume g: "finite {x. g x \<noteq> 0}" let ?f = "Max (insert 0 (Suc ` {k. f k \<noteq> 0}))" let ?g = "Max (insert 0 (Suc ` {k. g k \<noteq> 0}))" have "Max (insert 0 (Suc ` {k. f k + g k \<noteq> 0})) \<le> Max (insert 0 (Suc ` ({k. f k \<noteq> 0} \<union> {k. g k \<noteq> 0})))" by (rule Max.subset_imp) (insert f g, auto) also have "\<dots> = max ?f ?g" using f g by (simp_all add: image_Un Max_Un [symmetric]) finally show "Max (insert 0 (Suc ` {k. f k + g k \<noteq> 0})) \<le> max (Max (insert 0 (Suc ` {k. f k \<noteq> 0}))) (Max (insert 0 (Suc ` {k. g k \<noteq> 0})))" .qedlemma sorted_list_of_set_keys: "sorted_list_of_set (keys f) = filter (\<lambda>k. k \<in> keys f) [0..<degree f]" (is "_ = ?r")proof - have "keys f = set ?r" by (auto dest: in_keys_less_degree) moreover have "sorted_list_of_set (set ?r) = ?r" unfolding sorted_list_of_set_sort_remdups by (simp add: remdups_filter filter_sort [symmetric]) ultimately show ?thesis by simpqedsubsection \<open>Inductive structure\<close>lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b::zero) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b" is "\<lambda>k v f. f(k := v)"proof - fix f :: "'a \<Rightarrow> 'b" and k' v assume "finite {k. f k \<noteq> 0}" then have "finite (insert k' {k. f k \<noteq> 0})" by simp then show "finite {k. (f(k' := v)) k \<noteq> 0}" by (rule rev_finite_subset) autoqedlemma update_induct [case_names const update]: assumes const': "P 0" assumes update': "\<And>f a b. a \<notin> keys f \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> P f \<Longrightarrow> P (update a b f)" shows "P f"proof - obtain g where "f = Abs_poly_mapping g" and "finite {a. g a \<noteq> 0}" by (cases f) simp_all define Q where "Q g = P (Abs_poly_mapping g)" for g from \<open>finite {a. g a \<noteq> 0}\<close> have "Q g" proof (induct g rule: finite_update_induct) case const with const' Q_def show ?case by simp next case (update a b g) from \<open>finite {a. g a \<noteq> 0}\<close> \<open>g a = 0\<close> have "a \<notin> keys (Abs_poly_mapping g)" by (simp add: Abs_poly_mapping_inverse keys.rep_eq) moreover note \<open>b \<noteq> 0\<close> moreover from \<open>Q g\<close> have "P (Abs_poly_mapping g)" by (simp add: Q_def) ultimately have "P (update a b (Abs_poly_mapping g))" by (rule update') also from \<open>finite {a. g a \<noteq> 0}\<close> have "update a b (Abs_poly_mapping g) = Abs_poly_mapping (g(a := b))" by (simp add: update.abs_eq eq_onp_same_args) finally show ?case by (simp add: Q_def fun_upd_def) qed then show ?thesis by (simp add: Q_def \<open>f = Abs_poly_mapping g\<close>)qedlemma lookup_update: "lookup (update k v f) k' = (if k = k' then v else lookup f k')" by transfer simplemma keys_update: "keys (update k v f) = (if v = 0 then keys f - {k} else insert k (keys f))" by transfer autosubsection \<open>Quasi-functorial structure\<close>lift_definition map :: "('b::zero \<Rightarrow> 'c::zero) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c::zero)" is "\<lambda>g f k. g (f k) when f k \<noteq> 0" by simpcontext fixes f :: "'b \<Rightarrow> 'a" assumes inj_f: "inj f"beginlift_definition map_key :: "('a \<Rightarrow>\<^sub>0 'c::zero) \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'c" is "\<lambda>p. p \<circ> f"proof - fix g :: "'c \<Rightarrow> 'd" and p :: "'a \<Rightarrow> 'c" assume "finite {x. p x \<noteq> 0}" hence "finite (f ` {y. p (f y) \<noteq> 0})" by(rule finite_subset[rotated]) auto thus "finite {x. (p \<circ> f) x \<noteq> 0}" unfolding o_def by(rule finite_imageD)(rule subset_inj_on[OF inj_f], simp)qedendlemma map_key_compose: assumes [transfer_rule]: "inj f" "inj g" shows "map_key f (map_key g p) = map_key (g \<circ> f) p"proof - from assms have [transfer_rule]: "inj (g \<circ> f)" by(simp add: inj_compose) show ?thesis by transfer(simp add: o_assoc)qedlemma map_key_id: "map_key (\<lambda>x. x) p = p"proof - have [transfer_rule]: "inj (\<lambda>x. x)" by simp show ?thesis by transfer(simp add: o_def)qedcontext fixes f :: "'a \<Rightarrow> 'b" assumes inj_f [transfer_rule]: "inj f"beginlemma map_key_map: "map_key f (map g p) = map g (map_key f p)" by transfer (simp add: fun_eq_iff)lemma map_key_plus: "map_key f (p + q) = map_key f p + map_key f q" by transfer (simp add: fun_eq_iff)lemma keys_map_key: "keys (map_key f p) = f -` keys p" by transfer autolemma map_key_zero [simp]: "map_key f 0 = 0" by transfer (simp add: fun_eq_iff)lemma map_key_single [simp]: "map_key f (single (f k) v) = single k v" by transfer (simp add: fun_eq_iff inj_onD [OF inj_f] when_def)endlemma mult_map_scale_conv_mult: "map ((*) s) p = single 0 s * p"proof(transfer fixing: s) fix p :: "'a \<Rightarrow> 'b" assume *: "finite {x. p x \<noteq> 0}" { fix x have "prod_fun (\<lambda>k'. s when 0 = k') p x = (\<Sum>l :: 'a. if l = 0 then s * (\<Sum>q. p q when x = q) else 0)" by(auto simp add: prod_fun_def when_def intro: Sum_any.cong simp del: Sum_any.delta) also have "\<dots> = (\<lambda>k. s * p k when p k \<noteq> 0) x" by(simp add: when_def) also note calculation } then show "(\<lambda>k. s * p k when p k \<noteq> 0) = prod_fun (\<lambda>k'. s when 0 = k') p" by(simp add: fun_eq_iff)qedlemma map_single [simp]: "(c = 0 \<Longrightarrow> f 0 = 0) \<Longrightarrow> map f (single x c) = single x (f c)"by transfer(auto simp add: fun_eq_iff when_def)lemma map_eq_zero_iff: "map f p = 0 \<longleftrightarrow> (\<forall>k \<in> keys p. f (lookup p k) = 0)"by transfer(auto simp add: fun_eq_iff when_def)subsection \<open>Canonical dense representation of @{typ "nat \<Rightarrow>\<^sub>0 'a"}\<close>abbreviation no_trailing_zeros :: "'a :: zero list \<Rightarrow> bool"where "no_trailing_zeros \<equiv> no_trailing ((=) 0)"lift_definition "nth" :: "'a list \<Rightarrow> (nat \<Rightarrow>\<^sub>0 'a::zero)" is "nth_default 0" by (fact finite_nth_default_neq_default)text \<open> The opposite direction is directly specified on (later) type \<open>nat_mapping\<close>.\<close>lemma nth_Nil [simp]: "nth [] = 0" by transfer (simp add: fun_eq_iff)lemma nth_singleton [simp]: "nth [v] = single 0 v"proof (transfer, rule ext) fix n :: nat and v :: 'a show "nth_default 0 [v] n = (v when 0 = n)" by (auto simp add: nth_default_def nth_append)qedlemma nth_replicate [simp]: "nth (replicate n 0 @ [v]) = single n v"proof (transfer, rule ext) fix m n :: nat and v :: 'a show "nth_default 0 (replicate n 0 @ [v]) m = (v when n = m)" by (auto simp add: nth_default_def nth_append)qedlemma nth_strip_while [simp]: "nth (strip_while ((=) 0) xs) = nth xs" by transfer (fact nth_default_strip_while_dflt)lemma nth_strip_while' [simp]: "nth (strip_while (\<lambda>k. k = 0) xs) = nth xs" by (subst eq_commute) (fact nth_strip_while)lemma nth_eq_iff: "nth xs = nth ys \<longleftrightarrow> strip_while (HOL.eq 0) xs = strip_while (HOL.eq 0) ys" by transfer (simp add: nth_default_eq_iff)lemma lookup_nth [simp]: "lookup (nth xs) = nth_default 0 xs" by (fact nth.rep_eq)lemma keys_nth [simp]: "keys (nth xs) = fst ` {(n, v) \<in> set (enumerate 0 xs). v \<noteq> 0}"proof transfer fix xs :: "'a list" { fix n assume "nth_default 0 xs n \<noteq> 0" then have "n < length xs" and "xs ! n \<noteq> 0" by (auto simp add: nth_default_def split: if_splits) then have "(n, xs ! n) \<in> {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}" (is "?x \<in> ?A") by (auto simp add: in_set_conv_nth enumerate_eq_zip) then have "fst ?x \<in> fst ` ?A" by blast then have "n \<in> fst ` {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}" by simp } then show "{k. nth_default 0 xs k \<noteq> 0} = fst ` {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}" by (auto simp add: in_enumerate_iff_nth_default_eq)qedlemma range_nth [simp]: "range (nth xs) = set xs - {0}" by transfer simplemma degree_nth: "no_trailing_zeros xs \<Longrightarrow> degree (nth xs) = length xs"unfolding degree_def proof transfer fix xs :: "'a list" assume *: "no_trailing_zeros xs" let ?A = "{n. nth_default 0 xs n \<noteq> 0}" let ?f = "nth_default 0 xs" let ?bound = "Max (insert 0 (Suc ` {n. ?f n \<noteq> 0}))" show "?bound = length xs" proof (cases "xs = []") case False with * obtain n where n: "n < length xs" "xs ! n \<noteq> 0" by (fastforce simp add: no_trailing_unfold last_conv_nth neq_Nil_conv) then have "?bound = Max (Suc ` {k. (k < length xs \<longrightarrow> xs ! k \<noteq> (0::'a)) \<and> k < length xs})" by (subst Max_insert) (auto simp add: nth_default_def) also let ?A = "{k. k < length xs \<and> xs ! k \<noteq> 0}" have "{k. (k < length xs \<longrightarrow> xs ! k \<noteq> (0::'a)) \<and> k < length xs} = ?A" by auto also have "Max (Suc ` ?A) = Suc (Max ?A)" using n by (subst mono_Max_commute [where f = Suc, symmetric]) (auto simp add: mono_Suc) also { have "Max ?A \<in> ?A" using n Max_in [of ?A] by fastforce hence "Suc (Max ?A) \<le> length xs" by simp moreover from * False have "length xs - 1 \<in> ?A" by(auto simp add: no_trailing_unfold last_conv_nth) hence "length xs - 1 \<le> Max ?A" using Max_ge[of ?A "length xs - 1"] by auto hence "length xs \<le> Suc (Max ?A)" by simp ultimately have "Suc (Max ?A) = length xs" by simp } finally show ?thesis . qed simpqedlemma nth_trailing_zeros [simp]: "nth (xs @ replicate n 0) = nth xs" by transfer simplemma nth_idem: "nth (List.map (lookup f) [0..<degree f]) = f" unfolding degree_def by transfer (auto simp add: nth_default_def fun_eq_iff not_less)lemma nth_idem_bound: assumes "degree f \<le> n" shows "nth (List.map (lookup f) [0..<n]) = f"proof - from assms obtain m where "n = degree f + m" by (blast dest: le_Suc_ex) then have "[0..<n] = [0..<degree f] @ [degree f..<degree f + m]" by (simp add: upt_add_eq_append [of 0]) moreover have "List.map (lookup f) [degree f..<degree f + m] = replicate m 0" by (rule replicate_eqI) (auto simp add: beyond_degree_lookup_zero) ultimately show ?thesis by (simp add: nth_idem)qedsubsection \<open>Canonical sparse representation of @{typ "'a \<Rightarrow>\<^sub>0 'b"}\<close>lift_definition the_value :: "('a \<times> 'b) list \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b::zero" is "\<lambda>xs k. case map_of xs k of None \<Rightarrow> 0 | Some v \<Rightarrow> v"proof - fix xs :: "('a \<times> 'b) list" have fin: "finite {k. \<exists>v. map_of xs k = Some v}" using finite_dom_map_of [of xs] unfolding dom_def by auto then show "finite {k. (case map_of xs k of None \<Rightarrow> 0 | Some v \<Rightarrow> v) \<noteq> 0}" using fin by (simp split: option.split)qeddefinition items :: "('a::linorder \<Rightarrow>\<^sub>0 'b::zero) \<Rightarrow> ('a \<times> 'b) list"where "items f = List.map (\<lambda>k. (k, lookup f k)) (sorted_list_of_set (keys f))"text \<open> For the canonical sparse representation we provide both directions of morphisms since the specification of ordered association lists in theory \<open>OAList\<close> will support arbitrary linear orders @{class linorder} as keys, not just natural numbers @{typ nat}.\<close>lemma the_value_items [simp]: "the_value (items f) = f" unfolding items_def by transfer (simp add: fun_eq_iff map_of_map_restrict restrict_map_def)lemma lookup_the_value: "lookup (the_value xs) k = (case map_of xs k of None \<Rightarrow> 0 | Some v \<Rightarrow> v)" by transfer rulelemma items_the_value: assumes "sorted (List.map fst xs)" and "distinct (List.map fst xs)" and "0 \<notin> snd ` set xs" shows "items (the_value xs) = xs"proof - from assms have "sorted_list_of_set (set (List.map fst xs)) = List.map fst xs" unfolding sorted_list_of_set_sort_remdups by (simp add: distinct_remdups_id sort_key_id_if_sorted) moreover from assms have "keys (the_value xs) = fst ` set xs" by transfer (auto simp add: image_def split: option.split dest: set_map_of_compr) ultimately show ?thesis unfolding items_def using assms by (auto simp add: lookup_the_value intro: map_idI)qedlemma the_value_Nil [simp]: "the_value [] = 0" by transfer (simp add: fun_eq_iff)lemma the_value_Cons [simp]: "the_value (x # xs) = update (fst x) (snd x) (the_value xs)" by transfer (simp add: fun_eq_iff)lemma items_zero [simp]: "items 0 = []" unfolding items_def by simplemma items_one [simp]: "items 1 = [(0, 1)]" unfolding items_def by transfer simplemma items_single [simp]: "items (single k v) = (if v = 0 then [] else [(k, v)])" unfolding items_def by simplemma in_set_items_iff [simp]: "(k, v) \<in> set (items f) \<longleftrightarrow> k \<in> keys f \<and> lookup f k = v" unfolding items_def by transfer autosubsection \<open>Size estimation\<close>context fixes f :: "'a \<Rightarrow> nat" and g :: "'b :: zero \<Rightarrow> nat"begindefinition poly_mapping_size :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> nat"where "poly_mapping_size m = g 0 + (\<Sum>k \<in> keys m. Suc (f k + g (lookup m k)))"lemma poly_mapping_size_0 [simp]: "poly_mapping_size 0 = g 0" by (simp add: poly_mapping_size_def)lemma poly_mapping_size_single [simp]: "poly_mapping_size (single k v) = (if v = 0 then g 0 else g 0 + f k + g v + 1)" unfolding poly_mapping_size_def by transfer simplemma keys_less_poly_mapping_size: "k \<in> keys m \<Longrightarrow> f k + g (lookup m k) < poly_mapping_size m"unfolding poly_mapping_size_defproof transfer fix k :: 'a and m :: "'a \<Rightarrow> 'b" and f :: "'a \<Rightarrow> nat" and g let ?keys = "{k. m k \<noteq> 0}" assume *: "finite ?keys" "k \<in> ?keys" then have "f k + g (m k) = (\<Sum>k' \<in> ?keys. f k' + g (m k') when k' = k)" by (simp add: sum.delta when_def) also have "\<dots> < (\<Sum>k' \<in> ?keys. Suc (f k' + g (m k')))" using * by (intro sum_strict_mono) (auto simp add: when_def) also have "\<dots> \<le> g 0 + \<dots>" by simp finally have "f k + g (m k) < \<dots>" . then show "f k + g (m k) < g 0 + (\<Sum>k | m k \<noteq> 0. Suc (f k + g (m k)))" by simpqedlemma lookup_le_poly_mapping_size: "g (lookup m k) \<le> poly_mapping_size m"proof (cases "k \<in> keys m") case True with keys_less_poly_mapping_size [of k m] show ?thesis by simpnext case False then show ?thesis by (simp add: Poly_Mapping.poly_mapping_size_def in_keys_iff)qed lemma poly_mapping_size_estimation: "k \<in> keys m \<Longrightarrow> y \<le> f k + g (lookup m k) \<Longrightarrow> y < poly_mapping_size m" using keys_less_poly_mapping_size by (auto intro: le_less_trans)lemma poly_mapping_size_estimation2: assumes "v \<in> range m" and "y \<le> g v" shows "y < poly_mapping_size m"proof - from assms obtain k where *: "lookup m k = v" "v \<noteq> 0" by transfer blast from * have "k \<in> keys m" by (simp add: in_keys_iff) then show ?thesis proof (rule poly_mapping_size_estimation) from assms * have "y \<le> g (lookup m k)" by simp then show "y \<le> f k + g (lookup m k)" by simp qedqedendlemma poly_mapping_size_one [simp]: "poly_mapping_size f g 1 = g 0 + f 0 + g 1 + 1" unfolding poly_mapping_size_def by transfer simplemma poly_mapping_size_cong [fundef_cong]: "m = m' \<Longrightarrow> g 0 = g' 0 \<Longrightarrow> (\<And>k. k \<in> keys m' \<Longrightarrow> f k = f' k) \<Longrightarrow> (\<And>v. v \<in> range m' \<Longrightarrow> g v = g' v) \<Longrightarrow> poly_mapping_size f g m = poly_mapping_size f' g' m'" by (auto simp add: poly_mapping_size_def intro!: sum.cong)instantiation poly_mapping :: (type, zero) sizebegindefinition "size = poly_mapping_size (\<lambda>_. 0) (\<lambda>_. 0)"instance ..endsubsection \<open>Further mapping operations and properties\<close>text \<open>It is like in algebra: there are many definitions, some are also used\<close>lift_definition mapp :: "('a \<Rightarrow> 'b :: zero \<Rightarrow> 'c :: zero) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c)" is "\<lambda>f p k. (if k \<in> keys p then f k (lookup p k) else 0)" by simplemma mapp_cong [fundef_cong]: "\<lbrakk> m = m'; \<And>k. k \<in> keys m' \<Longrightarrow> f k (lookup m' k) = f' k (lookup m' k) \<rbrakk> \<Longrightarrow> mapp f m = mapp f' m'" by transfer (auto simp add: fun_eq_iff)lemma lookup_mapp: "lookup (mapp f p) k = (f k (lookup p k) when k \<in> keys p)" by (simp add: mapp.rep_eq)lemma keys_mapp_subset: "keys (mapp f p) \<subseteq> keys p" by (meson in_keys_iff mapp.rep_eq subsetI)subsection\<open>Free Abelian Groups Over a Type\<close>abbreviation frag_of :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>0 int" where "frag_of c \<equiv> Poly_Mapping.single c (1::int)"lemma lookup_frag_of [simp]: "Poly_Mapping.lookup(frag_of c) = (\<lambda>x. if x = c then 1 else 0)" by (force simp add: lookup_single_not_eq)lemma frag_of_nonzero [simp]: "frag_of a \<noteq> 0"proof - let ?f = "\<lambda>x. if x = a then 1 else (0::int)" have "?f \<noteq> (\<lambda>x. 0::int)" by (auto simp: fun_eq_iff) then have "Poly_Mapping.lookup (Abs_poly_mapping ?f) \<noteq> Poly_Mapping.lookup (Abs_poly_mapping (\<lambda>x. 0))" by fastforce then show ?thesis by (metis lookup_single_eq lookup_zero)qeddefinition frag_cmul :: "int \<Rightarrow> ('a \<Rightarrow>\<^sub>0 int) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 int)" where "frag_cmul c a = Abs_poly_mapping (\<lambda>x. c * Poly_Mapping.lookup a x)"lemma frag_cmul_zero [simp]: "frag_cmul 0 x = 0" by (simp add: frag_cmul_def)lemma frag_cmul_zero2 [simp]: "frag_cmul c 0 = 0" by (simp add: frag_cmul_def)lemma frag_cmul_one [simp]: "frag_cmul 1 x = x" by (auto simp: frag_cmul_def Poly_Mapping.poly_mapping.lookup_inverse)lemma frag_cmul_minus_one [simp]: "frag_cmul (-1) x = -x" by (simp add: frag_cmul_def uminus_poly_mapping_def poly_mapping_eqI)lemma frag_cmul_cmul [simp]: "frag_cmul c (frag_cmul d x) = frag_cmul (c*d) x" by (simp add: frag_cmul_def mult_ac)lemma lookup_frag_cmul [simp]: "poly_mapping.lookup (frag_cmul c x) i = c * poly_mapping.lookup x i" by (simp add: frag_cmul_def)lemma minus_frag_cmul [simp]: "- frag_cmul k x = frag_cmul (-k) x" by (simp add: poly_mapping_eqI)lemma keys_frag_of: "Poly_Mapping.keys(frag_of a) = {a}" by simplemma finite_cmul_nonzero: "finite {x. c * Poly_Mapping.lookup a x \<noteq> (0::int)}" by simplemma keys_cmul: "Poly_Mapping.keys(frag_cmul c a) \<subseteq> Poly_Mapping.keys a" using finite_cmul_nonzero [of c a] by (metis lookup_frag_cmul mult_zero_right not_in_keys_iff_lookup_eq_zero subsetI)lemma keys_cmul_iff [iff]: "i \<in> Poly_Mapping.keys (frag_cmul c x) \<longleftrightarrow> i \<in> Poly_Mapping.keys x \<and> c \<noteq> 0" apply auto apply (meson subsetD keys_cmul) by (metis in_keys_iff lookup_frag_cmul mult_eq_0_iff)lemma keys_minus [simp]: "Poly_Mapping.keys(-a) = Poly_Mapping.keys a" by (metis (no_types, opaque_lifting) in_keys_iff lookup_uminus neg_equal_0_iff_equal subsetI subset_antisym)lemma keys_diff: "Poly_Mapping.keys(a - b) \<subseteq> Poly_Mapping.keys a \<union> Poly_Mapping.keys b" by (auto simp add: in_keys_iff lookup_minus)lemma keys_eq_empty [simp]: "Poly_Mapping.keys c = {} \<longleftrightarrow> c = 0" by (metis in_keys_iff keys_zero lookup_zero poly_mapping_eqI)lemma frag_cmul_eq_0_iff [simp]: "frag_cmul k c = 0 \<longleftrightarrow> k=0 \<or> c=0" by auto (metis subsetI subset_antisym keys_cmul_iff keys_eq_empty)lemma frag_of_eq: "frag_of x = frag_of y \<longleftrightarrow> x = y" by (metis lookup_single_eq lookup_single_not_eq zero_neq_one)lemma frag_cmul_distrib: "frag_cmul (c+d) a = frag_cmul c a + frag_cmul d a" by (simp add: frag_cmul_def plus_poly_mapping_def int_distrib)lemma frag_cmul_distrib2: "frag_cmul c (a+b) = frag_cmul c a + frag_cmul c b"proof - have "finite {x. poly_mapping.lookup a x + poly_mapping.lookup b x \<noteq> 0}" using keys_add [of a b] by (metis (no_types, lifting) finite_keys finite_subset keys.rep_eq lookup_add mem_Collect_eq subsetI) then show ?thesis by (simp add: frag_cmul_def plus_poly_mapping_def int_distrib)qedlemma frag_cmul_diff_distrib: "frag_cmul (a - b) c = frag_cmul a c - frag_cmul b c" by (auto simp: left_diff_distrib lookup_minus poly_mapping_eqI)lemma frag_cmul_sum: "frag_cmul a (sum b I) = (\<Sum>i\<in>I. frag_cmul a (b i))"proof (induction rule: infinite_finite_induct) case (insert i I) then show ?case by (auto simp: algebra_simps frag_cmul_distrib2)qed autolemma keys_sum: "Poly_Mapping.keys(sum b I) \<subseteq> (\<Union>i \<in>I. Poly_Mapping.keys(b i))"proof (induction I rule: infinite_finite_induct) case (insert i I) then show ?case using keys_add [of "b i" "sum b I"] by autoqed autodefinition frag_extend :: "('b \<Rightarrow> 'a \<Rightarrow>\<^sub>0 int) \<Rightarrow> ('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 int" where "frag_extend b x \<equiv> (\<Sum>i \<in> Poly_Mapping.keys x. frag_cmul (Poly_Mapping.lookup x i) (b i))"lemma frag_extend_0 [simp]: "frag_extend b 0 = 0" by (simp add: frag_extend_def)lemma frag_extend_of [simp]: "frag_extend f (frag_of a) = f a" by (simp add: frag_extend_def)lemma frag_extend_cmul: "frag_extend f (frag_cmul c x) = frag_cmul c (frag_extend f x)" by (auto simp: frag_extend_def frag_cmul_sum intro: sum.mono_neutral_cong_left)lemma frag_extend_minus: "frag_extend f (- x) = - (frag_extend f x)" using frag_extend_cmul [of f "-1"] by simplemma frag_extend_add: "frag_extend f (a+b) = (frag_extend f a) + (frag_extend f b)"proof - have *: "(\<Sum>i\<in>Poly_Mapping.keys a. frag_cmul (poly_mapping.lookup a i) (f i)) = (\<Sum>i\<in>Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup a i) (f i))" "(\<Sum>i\<in>Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup b i) (f i)) = (\<Sum>i\<in>Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup b i) (f i))" by (auto simp: in_keys_iff intro: sum.mono_neutral_cong_left) have "frag_extend f (a+b) = (\<Sum>i\<in>Poly_Mapping.keys (a + b). frag_cmul (poly_mapping.lookup a i) (f i) + frag_cmul (poly_mapping.lookup b i) (f i)) " by (auto simp: frag_extend_def Poly_Mapping.lookup_add frag_cmul_distrib) also have "... = (\<Sum>i \<in> Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup a i) (f i) + frag_cmul (poly_mapping.lookup b i) (f i))" apply (rule sum.mono_neutral_cong_left) using keys_add [of a b] apply (auto simp add: in_keys_iff plus_poly_mapping.rep_eq frag_cmul_distrib [symmetric]) done also have "... = (frag_extend f a) + (frag_extend f b)" by (auto simp: * sum.distrib frag_extend_def) finally show ?thesis .qedlemma frag_extend_diff: "frag_extend f (a-b) = (frag_extend f a) - (frag_extend f b)" by (metis (no_types, opaque_lifting) add_uminus_conv_diff frag_extend_add frag_extend_minus)lemma frag_extend_sum: "finite I \<Longrightarrow> frag_extend f (\<Sum>i\<in>I. g i) = sum (frag_extend f o g) I" by (induction I rule: finite_induct) (simp_all add: frag_extend_add)lemma frag_extend_eq: "(\<And>f. f \<in> Poly_Mapping.keys c \<Longrightarrow> g f = h f) \<Longrightarrow> frag_extend g c = frag_extend h c" by (simp add: frag_extend_def)lemma frag_extend_eq_0: "(\<And>x. x \<in> Poly_Mapping.keys c \<Longrightarrow> f x = 0) \<Longrightarrow> frag_extend f c = 0" by (simp add: frag_extend_def)lemma keys_frag_extend: "Poly_Mapping.keys(frag_extend f c) \<subseteq> (\<Union>x \<in> Poly_Mapping.keys c. Poly_Mapping.keys(f x))" unfolding frag_extend_def using keys_sum by fastforcelemma frag_expansion: "a = frag_extend frag_of a"proof - have *: "finite I \<Longrightarrow> Poly_Mapping.lookup (\<Sum>i\<in>I. frag_cmul (Poly_Mapping.lookup a i) (frag_of i)) j = (if j \<in> I then Poly_Mapping.lookup a j else 0)" for I j by (induction I rule: finite_induct) (auto simp: lookup_single lookup_add) show ?thesis unfolding frag_extend_def by (rule poly_mapping_eqI) (fastforce simp add: in_keys_iff *)qedlemma frag_closure_minus_cmul: assumes "P 0" and P: "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> P(x - y)" "P c" shows "P(frag_cmul k c)"proof - have "P (frag_cmul (int n) c)" for n apply (induction n) apply (simp_all add: assms frag_cmul_distrib) by (metis add.left_neutral add_diff_cancel_right' add_uminus_conv_diff P) then show ?thesis by (metis (no_types, opaque_lifting) add_diff_eq assms(2) diff_add_cancel frag_cmul_distrib int_diff_cases)qedlemma frag_induction [consumes 1, case_names zero one diff]: assumes supp: "Poly_Mapping.keys c \<subseteq> S" and 0: "P 0" and sing: "\<And>x. x \<in> S \<Longrightarrow> P(frag_of x)" and diff: "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P(a - b)" shows "P c"proof - have "P (\<Sum>i\<in>I. frag_cmul (poly_mapping.lookup c i) (frag_of i))" if "I \<subseteq> Poly_Mapping.keys c" for I using finite_subset [OF that finite_keys [of c]] that supp proof (induction I arbitrary: c rule: finite_induct) case empty then show ?case by (auto simp: 0) next case (insert i I c) have ab: "a+b = a - (0 - b)" for a b :: "'a \<Rightarrow>\<^sub>0 int" by simp have Pfrag: "P (frag_cmul (poly_mapping.lookup c i) (frag_of i))" by (metis "0" diff frag_closure_minus_cmul insert.prems insert_subset sing subset_iff) show ?case apply (simp add: insert.hyps) apply (subst ab) using insert apply (blast intro: assms Pfrag) done qed then show ?thesis by (subst frag_expansion) (auto simp add: frag_extend_def)qedlemma frag_extend_compose: "frag_extend f (frag_extend (frag_of o g) c) = frag_extend (f o g) c" using subset_UNIV by (induction c rule: frag_induction) (auto simp: frag_extend_diff)lemma frag_split: fixes c :: "'a \<Rightarrow>\<^sub>0 int" assumes "Poly_Mapping.keys c \<subseteq> S \<union> T" obtains d e where "Poly_Mapping.keys d \<subseteq> S" "Poly_Mapping.keys e \<subseteq> T" "d + e = c"proof let ?d = "frag_extend (\<lambda>f. if f \<in> S then frag_of f else 0) c" let ?e = "frag_extend (\<lambda>f. if f \<in> S then 0 else frag_of f) c" show "Poly_Mapping.keys ?d \<subseteq> S" "Poly_Mapping.keys ?e \<subseteq> T" using assms by (auto intro!: order_trans [OF keys_frag_extend] split: if_split_asm) show "?d + ?e = c" using assms proof (induction c rule: frag_induction) case (diff a b) then show ?case by (metis (no_types, lifting) frag_extend_diff add_diff_eq diff_add_eq diff_add_eq_diff_diff_swap) qed autoqedhide_const (open) lookup single update keys range map map_key degree nth the_value items foldr mappend