(* This file is a part of IsarMathLib - a library of formalized mathematics for Isabelle/Isar. Copyright (C) 2007-2009 Slawomir Kolodynski This progr\rightarowam is free software; Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The name of the author may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. *) section ‹Semigroups› theory Semigroup_ZF imports Partitions_ZF Fold_ZF Enumeration_ZF begin text{*It seems that the minimal setup needed to talk about a product of a sequence is a set with a binary operation. Such object is called "magma". However, interesting properties show up when the binary operation is associative and such alebraic structure is called a semigroup. In this theory file we define and study sequences of partial products of sequences of magma and semigroup elements.*} subsection{*Products of sequences of semigroup elements*} text{*Semigroup is a a magma in which the binary operation is associative. In this section we mostly study the products of sequences of elements of semigroup. The goal is to establish the fact that taking the product of a sequence is distributive with respect to concatenation of sequences, i.e for two sequences $a,b$ of the semigroup elements we have $\prod (a\sqcup b) = (\prod a)\cdot (\prod b)$, where "$a \sqcup b$" is concatenation of $a$ and $b$ ($a$@{text "++"}$b$ in Haskell notation). Less formally, we want to show that we can discard parantheses in expressions of the form $(a_0\cdot a_1\cdot .. \cdot a_n)\cdot (b_0\cdot .. \cdot b_k)$. *} text{*First we define a notion similar to @{text "Fold"}, except that that the initial element of the fold is given by the first element of sequence. By analogy with Haskell fold we call that @{text "Fold1"} *} definition "Fold1(f,a) ≡ Fold(f,a`(0),Tail(a))" text{*The definition of the @{text "semigr0"} context below introduces notation for writing about finite sequences and semigroup products. In the context we fix the carrier and denote it $G$. The binary operation on $G$ is called $f$. All theorems proven in the context @{text "semigr0"} will implicitly assume that $f$ is an associative operation on $G$. We will use multiplicative notation for the semigroup operation. The product of a sequence $a$ is denoted $\prod a$. We will write $a\hookleftarrow x$ for the result of appending an element $x$ to the finite sequence (list) $a$. This is a bit nonstandard, but I don't have a better idea for the "append" notation. Finally, $a\sqcup b$ will denote the concatenation of the lists $a$ and $b$.*} locale semigr0 = fixes G f assumes assoc_assum: "f {is associative on} G" fixes prod (infixl "⋅" 72) defines prod_def [simp]: "x ⋅ y ≡ f`⟨x,y⟩" fixes seqprod ("∏ _" 71) defines seqprod_def [simp]: "∏ a ≡ Fold1(f,a)" fixes append (infix "↩" 72) defines append_def [simp]: "a ↩ x ≡ Append(a,x)" fixes concat (infixl "⊔" 69) defines concat_def [simp]: "a ⊔ b ≡ Concat(a,b)" text{*The next lemma shows our assumption on the associativity of the semigroup operation in the notation defined in in the @{text "semigr0"} context.*} lemma (in semigr0) semigr_assoc: assumes "x ∈ G" "y ∈ G" "z ∈ G" shows "x⋅y⋅z = x⋅(y⋅z)" using assms assoc_assum IsAssociative_def by simp text{*In the way we define associativity the assumption that $f$ is associative on $G$ also implies that it is a binary operation on $X$. *} lemma (in semigr0) semigr_binop: shows "f : G×G → G" using assoc_assum IsAssociative_def by simp text{*Semigroup operation is closed.*} lemma (in semigr0) semigr_closed: assumes "a∈G" "b∈G" shows "a⋅b ∈ G" using assms semigr_binop apply_funtype by simp text{*Lemma @{text "append_1elem"} written in the notation used in the @{text "semigr0"} context.*} lemma (in semigr0) append_1elem_nice: assumes "n ∈ nat" and "a: n → X" and "b : 1 → X" shows "a ⊔ b = a ↩ b`(0)" using assms append_1elem by simp text{*Lemma @{text "concat_init_last_elem"} rewritten in the notation used in the @{text "semigr0"} context.*} lemma (in semigr0) concat_init_last: assumes "n ∈ nat" "k ∈ nat" and "a: n → X" and "b : succ(k) → X" shows "(a ⊔ Init(b)) ↩ b`(k) = a ⊔ b" using assms concat_init_last_elem by simp text{*The product of semigroup (actually, magma -- we don't need associativity for this) elements is in the semigroup.*} lemma (in semigr0) prod_type: assumes "n ∈ nat" and "a : succ(n) → G" shows "(∏ a) ∈ G" proof - from assms have "succ(n) ∈ nat" "f : G×G → G" "Tail(a) : n → G" using semigr_binop tail_props by auto moreover from assms have "a`(0) ∈ G" and "G ≠ 0" using empty_in_every_succ apply_funtype by auto ultimately show "(∏ a) ∈ G" using Fold1_def fold_props by simp qed text{*What is the product of one element list?*} lemma (in semigr0) prod_of_1elem: assumes A1: "a: 1 → G" shows "(∏ a) = a`(0)" proof - have "f : G×G → G" using semigr_binop by simp moreover from A1 have "Tail(a) : 0 → G" using tail_props by blast moreover from A1 have "a`(0) ∈ G" and "G ≠ 0" using apply_funtype by auto ultimately show "(∏ a) = a`(0)" using fold_empty Fold1_def by simp qed text{*What happens to the product of a list when we append an element to the list?*} lemma (in semigr0) prod_append: assumes A1: "n ∈ nat" and A2: "a : succ(n) → G" and A3: "x∈G" shows "(∏ a↩x) = (∏ a) ⋅ x" proof - from A1 A2 have I: "Tail(a) : n → G" "a`(0) ∈ G" using tail_props empty_in_every_succ apply_funtype by auto from assms have "(∏ a↩x) = Fold(f,a`(0),Tail(a)↩x)" using head_of_append tail_append_commute Fold1_def by simp also from A1 A3 I have "… = (∏ a) ⋅ x" using semigr_binop fold_append Fold1_def by simp finally show ?thesis by simp qed text{*The main theorem of the section: taking the product of a sequence is distributive with respect to concatenation of sequences. The proof is by induction on the length of the second list.*} theorem (in semigr0) prod_conc_distr: assumes A1: "n ∈ nat" "k ∈ nat" and A2: "a : succ(n) → G" "b: succ(k) → G" shows "(∏ a) ⋅ (∏ b) = ∏ (a ⊔ b)" proof - from A1 have "k ∈ nat" by simp moreover have "∀b ∈ succ(0) → G. (∏ a) ⋅ (∏ b) = ∏ (a ⊔ b)" proof - { fix b assume A3: "b : succ(0) → G" with A1 A2 have "succ(n) ∈ nat" "a : succ(n) → G" "b : 1 → G" by auto then have "a ⊔ b = a ↩ b`(0)" by (rule append_1elem_nice) with A1 A2 A3 have "(∏ a) ⋅ (∏ b) = ∏ (a ⊔ b)" using apply_funtype prod_append semigr_binop prod_of_1elem by simp } thus ?thesis by simp qed moreover have "∀j ∈ nat. (∀b ∈ succ(j) → G. (∏ a) ⋅ (∏ b) = ∏ (a ⊔ b)) ⟶ (∀b ∈ succ(succ(j)) → G. (∏ a) ⋅ (∏ b) = ∏ (a ⊔ b))" proof - { fix j assume A4: "j ∈ nat" and A5: "(∀b ∈ succ(j) → G. (∏ a) ⋅ (∏ b) = ∏ (a ⊔ b))" { fix b assume A6: "b : succ(succ(j)) → G" let ?c = "Init(b)" from A4 A6 have T: "b`(succ(j)) ∈ G" and I: "?c : succ(j) → G" and II: "b = ?c↩b`(succ(j))" using apply_funtype init_props by auto from A1 A2 A4 A6 have "succ(n) ∈ nat" "succ(j) ∈ nat" "a : succ(n) → G" "b : succ(succ(j)) → G" by auto then have III: "(a ⊔ ?c) ↩ b`(succ(j)) = a ⊔ b" by (rule concat_init_last) from A4 I T have "(∏ ?c↩b`(succ(j))) = (∏ ?c) ⋅ b`(succ(j))" by (rule prod_append) with II have "(∏ a) ⋅ (∏ b) = (∏ a) ⋅ ((∏ ?c) ⋅ b`(succ(j)))" by simp moreover from A1 A2 A4 T I have "(∏ a) ∈ G" "(∏ ?c) ∈ G" "b`(succ(j)) ∈ G" using prod_type by auto ultimately have "(∏ a) ⋅ (∏ b) = ((∏ a) ⋅ (∏ ?c)) ⋅ b`(succ(j))" using semigr_assoc by auto with A5 I have "(∏ a) ⋅ (∏ b) = (∏ (a ⊔ ?c))⋅b`(succ(j))" by simp moreover from A1 A2 A4 I have T1: "succ(n) ∈ nat" "succ(j) ∈ nat" and "a : succ(n) → G" "?c : succ(j) → G" by auto then have "Concat(a,?c): succ(n) #+ succ(j) → G" by (rule concat_props) with A1 A4 T have "succ(n #+ j) ∈ nat" "a ⊔ ?c : succ(succ(n #+j)) → G" "b`(succ(j)) ∈ G" using succ_plus by auto then have "(∏ (a ⊔ ?c)↩b`(succ(j))) = (∏ (a ⊔ ?c))⋅b`(succ(j))" by (rule prod_append) with III have "(∏ (a ⊔ ?c))⋅b`(succ(j)) = ∏ (a ⊔ b)" by simp ultimately have "(∏ a) ⋅ (∏ b) = ∏ (a ⊔ b)" by simp } hence "(∀b ∈ succ(succ(j)) → G. (∏ a) ⋅ (∏ b) = ∏ (a ⊔ b))" by simp } thus ?thesis by blast qed ultimately have "∀b ∈ succ(k) → G. (∏ a) ⋅ (∏ b) = ∏ (a ⊔ b)" by (rule ind_on_nat) with A2 show "(∏ a) ⋅ (∏ b) = ∏ (a ⊔ b)" by simp qed subsection{*Products over sets of indices*} text{*In this section we study the properties of expressions of the form $\prod_{i\in \Lambda} a_i = a_{i_0}\cdot a_{i_1} \cdot .. \cdot a_{i-1}$, i.e. what we denote as @{text "\<pr>(Λ,a)"}. $\Lambda$ here is a finite subset of some set $X$ and $a$ is a function defined on $X$ with values in the semigroup $G$.*} text{* Suppose $a: X \rightarrow G$ is an indexed family of elements of a semigroup $G$ and $\Lambda = \{i_0, i_1, .. , i_{n-1}\} \subseteq \mathbb{N}$ is a finite set of indices. We want to define $\prod_{i\in \Lambda} a_i = a_{i_0}\cdot a_{i_1} \cdot .. \cdot a_{i-1}$. To do that we use the notion of @{text "Enumeration"} defined in the @{text "Enumeration_ZF"} theory file that takes a set of indices and lists them in increasing order, thus converting it to list. Then we use the @{text "Fold1"} to multiply the resulting list. Recall that in Isabelle/ZF the capital letter ''O'' denotes the composition of two functions (or relations). *} definition "SetFold(f,a,Λ,r) = Fold1(f,a O Enumeration(Λ,r))" text{* For a finite subset $\Lambda$ of a linearly ordered set $X$ we will write $\sigma (\Lambda )$ to denote the enumeration of the elements of $\Lambda$, i.e. the only order isomorphism $|\Lambda | \rightarrow \Lambda$, where $|\Lambda | \in \mathbb{N}$ is the number of elements of $\Lambda $. We also define notation for taking a product over a set of indices of some sequence of semigroup elements. The product of semigroup elements over some set $\Lambda \subseteq X$ of indices of a sequence $a: X \rightarrow G$ (i.e. $\prod_{i\in \Lambda} a_i$) is denoted @{text "\<pr>(Λ,a)"}. In the @{text "semigr1"} context we assume that $a$ is a function defined on some linearly ordered set $X$ with values in the semigroup $G$. *} locale semigr1 = semigr0 + fixes X r assumes linord: "IsLinOrder(X,r)" fixes a assumes a_is_fun: "a : X → G" fixes σ defines σ_def [simp]: "σ(A) ≡ Enumeration(A,r)" fixes setpr ("\<pr>") defines setpr_def [simp]: "\<pr>(Λ,b) ≡ SetFold(f,b,Λ,r)" text{*We can use the @{text "enums"} locale in the @{text "semigr0"} context.*} lemma (in semigr1) enums_valid_in_semigr1: shows "enums(X,r)" using linord enums_def by simp text{*Definition of product over a set expressed in notation of the @{text "semigr0"} locale.*} lemma (in semigr1) setproddef: shows "\<pr>(Λ,a) = ∏ (a O σ(Λ))" using SetFold_def by simp text{*A composition of enumeration of a nonempty finite subset of $\mathbb{N}$ with a sequence of elements of $G$ is a nonempty list of elements of $G$. This implies that a product over set of a finite set of indices belongs to the (carrier of) semigroup. *} lemma (in semigr1) setprod_type: assumes A1: "Λ ∈ FinPow(X)" and A2: "Λ≠0" shows "∃n ∈ nat . |Λ| = succ(n) ∧ a O σ(Λ) : succ(n) → G" and "\<pr>(Λ,a) ∈ G" proof - from assms obtain n where "n ∈ nat" and "|Λ| = succ(n)" using card_non_empty_succ by auto from A1 have "σ(Λ) : |Λ| → Λ" using enums_valid_in_semigr1 enums.enum_props by simp with A1 have "a O σ(Λ): |Λ| → G" using a_is_fun FinPow_def comp_fun_subset by simp with `n ∈ nat` and `|Λ| = succ(n)` show "∃n ∈ nat . |Λ| = succ(n) ∧ a O σ(Λ) : succ(n) → G" by auto from `n ∈ nat` `|Λ| = succ(n)` `a O σ(Λ): |Λ| → G` show "\<pr>(Λ,a) ∈ G" using prod_type setproddef by auto qed text{*The @{text "enum_append"} lemma from the @{text "Enemeration"} theory specialized for natural numbers.*} lemma (in semigr1) semigr1_enum_append: assumes "Λ ∈ FinPow(X)" and "n ∈ X - Λ" and "∀k∈Λ. ⟨k,n⟩ ∈ r" shows "σ(Λ ∪ {n}) = σ(Λ)↩ n" using assms FinPow_def enums_valid_in_semigr1 enums.enum_append by simp text{*What is product over a singleton?*} lemma (in semigr1) gen_prod_singleton: assumes A1: "x ∈ X" shows "\<pr>({x},a) = a`(x)" proof - from A1 have "σ({x}): 1 → X" and "σ({x})`(0) = x" using enums_valid_in_semigr1 enums.enum_singleton by auto then show "\<pr>({x},a) = a`(x)" using a_is_fun comp_fun setproddef prod_of_1elem comp_fun_apply by simp qed text{*A generalization of @{text "prod_append"} to the products over sets of indices.*} lemma (in semigr1) gen_prod_append: assumes A1: "Λ ∈ FinPow(X)" and A2: "Λ ≠ 0" and A3: "n ∈ X - Λ" and A4: "∀k∈Λ. ⟨k,n⟩ ∈ r" shows "\<pr>(Λ ∪ {n}, a) = (\<pr>(Λ,a)) ⋅ a`(n)" proof - have "\<pr>(Λ ∪ {n}, a) = ∏ (a O σ(Λ ∪ {n}))" using setproddef by simp also from A1 A3 A4 have "… = ∏ (a O (σ(Λ)↩ n))" using semigr1_enum_append by simp also have "… = ∏ ((a O σ(Λ))↩ a`(n))" proof - from A1 A3 have "|Λ| ∈ nat" and "σ(Λ) : |Λ| → X" and "n ∈ X" using card_fin_is_nat enums_valid_in_semigr1 enums.enum_fun by auto then show ?thesis using a_is_fun list_compose_append by simp qed also from assms have "… = (∏ (a O σ(Λ)))⋅a`(n)" using a_is_fun setprod_type apply_funtype prod_append by blast also have "… = (\<pr>(Λ,a)) ⋅ a`(n)" using SetFold_def by simp finally show "\<pr>(Λ ∪ {n}, a) = (\<pr>(Λ,a)) ⋅ a`(n)" by simp qed text{*Very similar to @{text "gen_prod_append"}: a relation between a product over a set of indices and the product over the set with the maximum removed. *} lemma (in semigr1) gen_product_rem_point: assumes A1: "A ∈ FinPow(X)" and A2: "n ∈ A" and A4: "A - {n} ≠ 0" and A3: "∀k∈A. ⟨k, n⟩ ∈ r" shows "(\<pr>(A - {n},a)) ⋅ a`(n) = \<pr>(A, a)" proof - let ?Λ = "A - {n}" from A1 A2 have "?Λ ∈ FinPow(X)" and "n ∈ X - ?Λ" using fin_rem_point_fin FinPow_def by auto with A3 A4 have "\<pr>(?Λ ∪ {n}, a) = (\<pr>(?Λ,a)) ⋅ a`(n)" using a_is_fun gen_prod_append by blast with A2 show ?thesis using rem_add_eq by simp qed subsection{*Commutative semigroups*} text{*Commutative semigroups are those whose operation is commutative, i.e. $a\cdot b = b\cdot a$. This implies that for any permutation $s : n \rightarrow n$ we have $\prod_{j=0}^n a_j = \prod_{j=0}^n a_{s (j)}$, or, closer to the notation we are using in the @{text "semigr0"} context, $\prod a = \prod (a \circ s )$. Maybe one day we will be able to prove this, but for now the goal is to prove something simpler: that if the semigroup operation is commutative taking the product of a sequence is distributive with respect to the operation: $\prod_{j=0}^n (a_j\cdot b_j) = \left(\prod_{j=0}^n a_j)\right) \left(\prod_{j=0}^n b_j)\right)$. Many of the rearrangements (namely those that don't use the inverse) proven in the @{text "AbelianGroup_ZF"} theory hold in fact in semigroups. Some of them will be reproven in this section.*} text{*A rearrangement with 3 elements.*} lemma (in semigr0) rearr3elems: assumes "f {is commutative on} G" and "a∈G" "b∈G" "c∈G" shows "a⋅b⋅c = a⋅c⋅b" using assms semigr_assoc IsCommutative_def by simp text{*A rearrangement of four elements.*} lemma (in semigr0) rearr4elems: assumes A1: "f {is commutative on} G" and A2: "a∈G" "b∈G" "c∈G" "d∈G" shows "a⋅b⋅(c⋅d) = a⋅c⋅(b⋅d)" proof - from A2 have "a⋅b⋅(c⋅d) = a⋅b⋅c⋅d" using semigr_closed semigr_assoc by simp also have "a⋅b⋅c⋅d = a⋅c⋅(b⋅d)" proof - from A1 A2 have "a⋅b⋅c⋅d = c⋅(a⋅b)⋅d" using IsCommutative_def semigr_closed by simp also from A2 have "… = c⋅a⋅b⋅d" using semigr_closed semigr_assoc by simp also from A1 A2 have "… = a⋅c⋅b⋅d" using IsCommutative_def semigr_closed by simp also from A2 have "… = a⋅c⋅(b⋅d)" using semigr_closed semigr_assoc by simp finally show "a⋅b⋅c⋅d = a⋅c⋅(b⋅d)" by simp qed finally show "a⋅b⋅(c⋅d) = a⋅c⋅(b⋅d)" by simp qed text{*We start with a version of @{text "prod_append"} that will shorten a bit the proof of the main theorem.*} lemma (in semigr0) shorter_seq: assumes A1: "k ∈ nat" and A2: "a ∈ succ(succ(k)) → G" shows "(∏ a) = (∏ Init(a)) ⋅ a`(succ(k))" proof - let ?x = "Init(a)" from assms have "a`(succ(k)) ∈ G" and "?x : succ(k) → G" using apply_funtype init_props by auto with A1 have "(∏ ?x↩a`(succ(k))) = (∏ ?x) ⋅ a`(succ(k))" using prod_append by simp with assms show ?thesis using init_props by simp qed text{*A lemma useful in the induction step of the main theorem.*} lemma (in semigr0) prod_distr_ind_step: assumes A1: "k ∈ nat" and A2: "a : succ(succ(k)) → G" and A3: "b : succ(succ(k)) → G" and A4: "c : succ(succ(k)) → G" and A5: "∀j∈succ(succ(k)). c`(j) = a`(j) ⋅ b`(j)" shows "Init(a) : succ(k) → G" "Init(b) : succ(k) → G" "Init(c) : succ(k) → G" "∀j∈succ(k). Init(c)`(j) = Init(a)`(j) ⋅ Init(b)`(j)" proof - from A1 A2 A3 A4 show "Init(a) : succ(k) → G" "Init(b) : succ(k) → G" "Init(c) : succ(k) → G" using init_props by auto from A1 have T: "succ(k) ∈ nat" by simp from T A2 have "∀j∈succ(k). Init(a)`(j) = a`(j)" by (rule init_props) moreover from T A3 have "∀j∈succ(k). Init(b)`(j) = b`(j)" by (rule init_props) moreover from T A4 have "∀j∈succ(k). Init(c)`(j) = c`(j)" by (rule init_props) moreover from A5 have "∀j∈succ(k). c`(j) = a`(j) ⋅ b`(j)" by simp ultimately show "∀j∈succ(k). Init(c)`(j) = Init(a)`(j) ⋅ Init(b)`(j)" by simp qed text{*For commutative operations taking the product of a sequence is distributive with respect to the operation. This version will probably not be used in applications, it is formulated in a way that is easier to prove by induction. For a more convenient formulation see @{text "prod_comm_distrib"}. The proof by induction on the length of the sequence.*} theorem (in semigr0) prod_comm_distr: assumes A1: "f {is commutative on} G" and A2: "n∈nat" shows "∀ a b c. (a : succ(n)→G ∧ b : succ(n)→G ∧ c : succ(n)→G ∧ (∀j∈succ(n). c`(j) = a`(j) ⋅ b`(j))) ⟶ (∏ c) = (∏ a) ⋅ (∏ b)" proof - note A2 moreover have "∀ a b c. (a : succ(0)→G ∧ b : succ(0)→G ∧ c : succ(0)→G ∧ (∀j∈succ(0). c`(j) = a`(j) ⋅ b`(j))) ⟶ (∏ c) = (∏ a) ⋅ (∏ b)" proof - { fix a b c assume "a : succ(0)→G ∧ b : succ(0)→G ∧ c : succ(0)→G ∧ (∀j∈succ(0). c`(j) = a`(j) ⋅ b`(j))" then have I: "a : 1→G" "b : 1→G" "c : 1→G" and II: "c`(0) = a`(0) ⋅ b`(0)" by auto from I have "(∏ a) = a`(0)" and "(∏ b) = b`(0)" and "(∏ c) = c`(0)" using prod_of_1elem by auto with II have "(∏ c) = (∏ a) ⋅ (∏ b)" by simp } then show ?thesis using Fold1_def by simp qed moreover have "∀k ∈ nat. (∀ a b c. (a : succ(k)→G ∧ b : succ(k)→G ∧ c : succ(k)→G ∧ (∀j∈succ(k). c`(j) = a`(j) ⋅ b`(j))) ⟶ (∏ c) = (∏ a) ⋅ (∏ b)) ⟶ (∀ a b c. (a : succ(succ(k))→G ∧ b : succ(succ(k))→G ∧ c : succ(succ(k))→G ∧ (∀j∈succ(succ(k)). c`(j) = a`(j) ⋅ b`(j))) ⟶ (∏ c) = (∏ a) ⋅ (∏ b))" proof fix k assume "k ∈ nat" show "(∀a b c. a ∈ succ(k) → G ∧ b ∈ succ(k) → G ∧ c ∈ succ(k) → G ∧ (∀j∈succ(k). c`(j) = a`(j) ⋅ b`(j)) ⟶ (∏ c) = (∏ a) ⋅ (∏ b)) ⟶ (∀a b c. a ∈ succ(succ(k)) → G ∧ b ∈ succ(succ(k)) → G ∧ c ∈ succ(succ(k)) → G ∧ (∀j∈succ(succ(k)). c`(j) = a`(j) ⋅ b`(j)) ⟶ (∏ c) = (∏ a) ⋅ (∏ b))" proof assume A3: "∀a b c. a ∈ succ(k) → G ∧ b ∈ succ(k) → G ∧ c ∈ succ(k) → G ∧ (∀j∈succ(k). c`(j) = a`(j) ⋅ b`(j)) ⟶ (∏ c) = (∏ a) ⋅ (∏ b)" show "∀a b c. a ∈ succ(succ(k)) → G ∧ b ∈ succ(succ(k)) → G ∧ c ∈ succ(succ(k)) → G ∧ (∀j∈succ(succ(k)). c`(j) = a`(j) ⋅ b`(j)) ⟶ (∏ c) = (∏ a) ⋅ (∏ b)" proof - { fix a b c assume "a ∈ succ(succ(k)) → G ∧ b ∈ succ(succ(k)) → G ∧ c ∈ succ(succ(k)) → G ∧ (∀j∈succ(succ(k)). c`(j) = a`(j) ⋅ b`(j))" with `k ∈ nat` have I: "a : succ(succ(k)) → G" "b : succ(succ(k)) → G" "c : succ(succ(k)) → G" and II: "∀j∈succ(succ(k)). c`(j) = a`(j) ⋅ b`(j)" by auto let ?x = "Init(a)" let ?y = "Init(b)" let ?z = "Init(c)" from `k ∈ nat` I have III: "(∏ a) = (∏ ?x) ⋅ a`(succ(k))" "(∏ b) = (∏ ?y) ⋅ b`(succ(k))" and IV: "(∏ c) = (∏ ?z) ⋅ c`(succ(k))" using shorter_seq by auto moreover from `k ∈ nat` I II have "?x : succ(k) → G" "?y : succ(k) → G" "?z : succ(k) → G" and "∀j∈succ(k). ?z`(j) = ?x`(j) ⋅ ?y`(j)" using prod_distr_ind_step by auto with A3 II IV have "(∏ c) = (∏ ?x)⋅(∏ ?y)⋅(a`(succ(k)) ⋅ b`(succ(k)))" by simp moreover from A1 `k ∈ nat` I III have "(∏ ?x)⋅(∏ ?y)⋅(a`(succ(k)) ⋅ b`(succ(k)))= (∏ a) ⋅ (∏ b)" using init_props prod_type apply_funtype rearr4elems by simp ultimately have "(∏ c) = (∏ a) ⋅ (∏ b)" by simp } thus ?thesis by auto qed qed qed ultimately show ?thesis by (rule ind_on_nat) qed text{*A reformulation of @{text "prod_comm_distr"} that is more convenient in applications.*} theorem (in semigr0) prod_comm_distrib: assumes "f {is commutative on} G" and "n∈nat" and "a : succ(n)→G" "b : succ(n)→G" "c : succ(n)→G" and "∀j∈succ(n). c`(j) = a`(j) ⋅ b`(j)" shows "(∏ c) = (∏ a) ⋅ (∏ b)" using assms prod_comm_distr by simp text{*A product of two products over disjoint sets of indices is the product over the union.*} lemma (in semigr1) prod_bisect: assumes A1: "f {is commutative on} G" and A2: "Λ ∈ FinPow(X)" shows "∀P ∈ Bisections(Λ). \<pr>(Λ,a) = (\<pr>(fst(P),a))⋅(\<pr>(snd(P),a))" proof - have "IsLinOrder(X,r)" using linord by simp moreover have "∀P ∈ Bisections(0). \<pr>(0,a) = (\<pr>(fst(P),a))⋅(\<pr>(snd(P),a))" using bisec_empty by simp moreover have "∀ A ∈ FinPow(X). ( ∀ n ∈ X - A. (∀P ∈ Bisections(A). \<pr>(A,a) = (\<pr>(fst(P),a))⋅(\<pr>(snd(P),a))) ∧ (∀k∈A. ⟨k,n⟩ ∈ r ) ⟶ (∀Q ∈ Bisections(A ∪ {n}). \<pr>(A ∪ {n},a) = (\<pr>(fst(Q),a))⋅(\<pr>(snd(Q),a))))" proof - { fix A assume "A ∈ FinPow(X)" fix n assume "n ∈ X - A" have "( ∀P ∈ Bisections(A). \<pr>(A,a) = (\<pr>(fst(P),a))⋅(\<pr>(snd(P),a))) ∧ (∀k∈A. ⟨k,n⟩ ∈ r ) ⟶ (∀Q ∈ Bisections(A ∪ {n}). \<pr>(A ∪ {n},a) = (\<pr>(fst(Q),a))⋅(\<pr>(snd(Q),a)))" proof - { assume I: "∀P ∈ Bisections(A). \<pr>(A,a) = (\<pr>(fst(P),a))⋅(\<pr>(snd(P),a))" and II: "∀k∈A. ⟨k,n⟩ ∈ r" have "∀Q ∈ Bisections(A ∪ {n}). \<pr>(A ∪ {n},a) = (\<pr>(fst(Q),a))⋅(\<pr>(snd(Q),a))" proof - { fix Q assume "Q ∈ Bisections(A ∪ {n})" let ?Q⇩_{0}= "fst(Q)" let ?Q⇩_{1}= "snd(Q)" from `A ∈ FinPow(X)` `n ∈ X - A` have "A ∪ {n} ∈ FinPow(X)" using singleton_in_finpow union_finpow by auto with `Q ∈ Bisections(A ∪ {n})` have "?Q⇩_{0}∈ FinPow(X)" "?Q⇩_{0}≠ 0" and "?Q⇩_{1}∈ FinPow(X)" "?Q⇩_{1}≠ 0" using bisect_fin bisec_is_pair Bisections_def by auto then have "\<pr>(?Q⇩_{0},a) ∈ G" and "\<pr>(?Q⇩_{1},a) ∈ G" using a_is_fun setprod_type by auto from `Q ∈ Bisections(A ∪ {n})` `A ∈ FinPow(X)` `n ∈ X-A` have "refl(X,r)" "?Q⇩_{0}⊆ A ∪ {n}" "?Q⇩_{1}⊆ A ∪ {n}" "A ⊆ X" and "n ∈ X" using linord IsLinOrder_def total_is_refl Bisections_def FinPow_def by auto from `refl(X,r)` `?Q⇩_{0}⊆ A ∪ {n}` `A ⊆ X` `n ∈ X` II have III: "∀k ∈ ?Q⇩_{0}. ⟨k, n⟩ ∈ r" by (rule refl_add_point) from `refl(X,r)` `?Q⇩_{1}⊆ A ∪ {n}` `A ⊆ X` `n ∈ X` II have IV: "∀k ∈ ?Q⇩_{1}. ⟨k, n⟩ ∈ r" by (rule refl_add_point) from `n ∈ X - A` `Q ∈ Bisections(A ∪ {n})` have "?Q⇩_{0}= {n} ∨ ?Q⇩_{1}= {n} ∨ ⟨?Q⇩_{0}- {n},?Q⇩_{1}-{n}⟩ ∈ Bisections(A)" using bisec_is_pair bisec_add_point by simp moreover { assume "?Q⇩_{1}= {n}" from `n ∈ X - A` have "n ∉ A" by auto moreover from `Q ∈ Bisections(A ∪ {n})` have "⟨?Q⇩_{0},?Q⇩_{1}⟩ ∈ Bisections(A ∪ {n})" using bisec_is_pair by simp with `?Q⇩_{1}= {n}` have "⟨?Q⇩_{0}, {n}⟩ ∈ Bisections(A ∪ {n})" by simp ultimately have "?Q⇩_{0}= A" and "A ≠ 0" using set_point_bisec by auto with `A ∈ FinPow(X)` `n ∈ X - A` II `?Q⇩_{1}= {n}` have "\<pr>(A ∪ {n},a) = (\<pr>(?Q⇩_{0},a))⋅\<pr>(?Q⇩_{1},a)" using a_is_fun gen_prod_append gen_prod_singleton by simp } moreover { assume "?Q⇩_{0}= {n}" from `n ∈ X - A` have "n ∈ X" by auto then have "{n} ∈ FinPow(X)" and "{n} ≠ 0" using singleton_in_finpow by auto from `n ∈ X - A` have "n ∉ A" by auto moreover from `Q ∈ Bisections(A ∪ {n})` have "⟨?Q⇩_{0}, ?Q⇩_{1}⟩ ∈ Bisections(A ∪ {n})" using bisec_is_pair by simp with `?Q⇩_{0}= {n}` have "⟨{n}, ?Q⇩_{1}⟩ ∈ Bisections(A ∪ {n})" by simp ultimately have "?Q⇩_{1}= A" and "A ≠ 0" using point_set_bisec by auto with A1 `A ∈ FinPow(X)` `n ∈ X - A` II `{n} ∈ FinPow(X)` `{n} ≠ 0` `?Q⇩_{0}= {n}` have "\<pr>(A ∪ {n},a) = (\<pr>(?Q⇩_{0},a))⋅(\<pr>(?Q⇩_{1},a))" using a_is_fun gen_prod_append gen_prod_singleton setprod_type IsCommutative_def by auto } moreover { assume A4: "⟨?Q⇩_{0}- {n},?Q⇩_{1}- {n}⟩ ∈ Bisections(A)" with `A ∈ FinPow(X)` have "?Q⇩_{0}- {n} ∈ FinPow(X)" "?Q⇩_{0}- {n} ≠ 0" and "?Q⇩_{1}- {n} ∈ FinPow(X)" "?Q⇩_{1}- {n} ≠ 0" using FinPow_def Bisections_def by auto with `n ∈ X - A` have "\<pr>(?Q⇩_{0}- {n},a) ∈ G" "\<pr>(?Q⇩_{1}- {n},a) ∈ G" and T: "a`(n) ∈ G" using a_is_fun setprod_type apply_funtype by auto from `Q ∈ Bisections(A ∪ {n})` A4 have "(⟨?Q⇩_{0}, ?Q⇩_{1}- {n}⟩ ∈ Bisections(A) ∧ n ∈ ?Q⇩_{1}) ∨ (⟨?Q⇩_{0}- {n}, ?Q⇩_{1}⟩ ∈ Bisections(A) ∧ n ∈ ?Q⇩_{0}) " using bisec_is_pair bisec_add_point_case3 by auto moreover { assume "⟨?Q⇩_{0}, ?Q⇩_{1}- {n}⟩ ∈ Bisections(A)" and "n ∈ ?Q⇩_{1}" then have "A ≠ 0" using bisec_props by simp with A2 `A ∈ FinPow(X)` `n ∈ X - A` I II T IV `⟨?Q⇩_{0}, ?Q⇩_{1}- {n}⟩ ∈ Bisections(A)` `\<pr>(?Q⇩_{0},a) ∈ G` `\<pr>(?Q⇩_{1}- {n},a) ∈ G` `?Q⇩_{1}∈ FinPow(X)` `n ∈ ?Q⇩_{1}` `?Q⇩_{1}- {n} ≠ 0` have "\<pr>(A ∪ {n},a) = (\<pr>(?Q⇩_{0},a))⋅(\<pr>(?Q⇩_{1},a))" using gen_prod_append semigr_assoc gen_product_rem_point by simp } moreover { assume "⟨?Q⇩_{0}- {n}, ?Q⇩_{1}⟩ ∈ Bisections(A)" and "n ∈ ?Q⇩_{0}" then have "A ≠ 0" using bisec_props by simp with A1 A2 `A ∈ FinPow(X)` `n ∈ X - A` I II III T `⟨?Q⇩_{0}- {n}, ?Q⇩_{1}⟩∈Bisections(A)` `\<pr>(?Q⇩_{0}- {n},a)∈G` `\<pr>(?Q⇩_{1},a) ∈ G` `?Q⇩_{0}∈ FinPow(X)` `n ∈ ?Q⇩_{0}` `?Q⇩_{0}-{n}≠0` have "\<pr>(A ∪ {n},a) = (\<pr>(?Q⇩_{0},a))⋅(\<pr>(?Q⇩_{1},a))" using gen_prod_append rearr3elems gen_product_rem_point by simp } ultimately have "\<pr>(A ∪ {n},a) = (\<pr>(?Q⇩_{0},a))⋅(\<pr>(?Q⇩_{1},a))" by auto } ultimately have "\<pr>(A ∪ {n},a) = (\<pr>(?Q⇩_{0},a))⋅(\<pr>(?Q⇩_{1},a))" by auto } thus ?thesis by simp qed } thus ?thesis by simp qed } thus ?thesis by simp qed moreover note A2 ultimately show ?thesis by (rule fin_ind_add_max) qed text{*A better looking reformulation of @{text "prod_bisect"}. *} theorem (in semigr1) prod_disjoint: assumes A1: "f {is commutative on} G" and A2: "A ∈ FinPow(X)" "A ≠ 0" and A3: "B ∈ FinPow(X)" "B ≠ 0" and A4: "A ∩ B = 0" shows "\<pr>(A∪B,a) = (\<pr>(A,a))⋅(\<pr>(B,a))" proof - from A2 A3 A4 have "⟨A,B⟩ ∈ Bisections(A∪B)" using is_bisec by simp with A1 A2 A3 show ?thesis using a_is_fun union_finpow prod_bisect by simp qed text{*A generalization of @{text "prod_disjoint"}.*} lemma (in semigr1) prod_list_of_lists: assumes A1: "f {is commutative on} G" and A2: "n ∈ nat" shows "∀M ∈ succ(n) → FinPow(X). M {is partition} ⟶ (∏ {⟨i,\<pr>(M`(i),a)⟩. i ∈ succ(n)}) = (\<pr>(⋃i ∈ succ(n). M`(i),a))" proof - note A2 moreover have "∀M ∈ succ(0) → FinPow(X). M {is partition} ⟶ (∏ {⟨i,\<pr>(M`(i),a)⟩. i ∈ succ(0)}) = (\<pr>(⋃i ∈ succ(0). M`(i),a))" using a_is_fun func1_1_L1 Partition_def apply_funtype setprod_type list_len1_singleton prod_of_1elem by simp moreover have "∀k ∈ nat. (∀M ∈ succ(k) → FinPow(X). M {is partition} ⟶ (∏ {⟨i,\<pr>(M`(i),a)⟩. i ∈ succ(k)}) = (\<pr>(⋃i ∈ succ(k). M`(i),a))) ⟶ (∀M ∈ succ(succ(k)) → FinPow(X). M {is partition} ⟶ (∏ {⟨i,\<pr>(M`(i),a)⟩. i ∈ succ(succ(k))}) = (\<pr>(⋃i ∈ succ(succ(k)). M`(i),a)))" proof - { fix k assume "k ∈ nat" assume A3: "∀M ∈ succ(k) → FinPow(X). M {is partition} ⟶ (∏ {⟨i,\<pr>(M`(i),a)⟩. i ∈ succ(k)}) = (\<pr>(⋃i ∈ succ(k). M`(i),a))" have "(∀N ∈ succ(succ(k)) → FinPow(X). N {is partition} ⟶ (∏ {⟨i,\<pr>(N`(i),a)⟩. i ∈ succ(succ(k))}) = (\<pr>(⋃i ∈ succ(succ(k)). N`(i),a)))" proof - { fix N assume A4: "N : succ(succ(k)) → FinPow(X)" assume A5: "N {is partition}" with A4 have I: "∀i ∈ succ(succ(k)). N`(i) ≠ 0" using func1_1_L1 Partition_def by simp let ?b = "{⟨i,\<pr>(N`(i),a)⟩. i ∈ succ(succ(k))}" let ?c = "{⟨i,\<pr>(N`(i),a)⟩. i ∈ succ(k)}" have II: "∀i ∈ succ(succ(k)). \<pr>(N`(i),a) ∈ G" proof fix i assume "i ∈ succ(succ(k))" with A4 I have "N`(i) ∈ FinPow(X)" and "N`(i) ≠ 0" using apply_funtype by auto then show "\<pr>(N`(i),a) ∈ G" using setprod_type by simp qed hence "∀i ∈ succ(k). \<pr>(N`(i),a) ∈ G" by auto then have "?c : succ(k) → G" by (rule ZF_fun_from_total) have "?b = {⟨i,\<pr>(N`(i),a)⟩. i ∈ succ(succ(k))}" by simp with II have "?b = Append(?c,\<pr>(N`(succ(k)),a))" by (rule set_list_append) with II `k ∈ nat` `?c : succ(k) → G` have "(∏ ?b) = (∏ ?c)⋅(\<pr>(N`(succ(k)),a))" using prod_append by simp also have "… = (\<pr>(⋃i ∈ succ(k). N`(i),a))⋅(\<pr>(N`(succ(k)),a))" proof - let ?M = "restrict(N,succ(k))" have "succ(k) ⊆ succ(succ(k))" by auto with `N : succ(succ(k)) → FinPow(X)` have "?M : succ(k) → FinPow(X)" and III: "∀i ∈ succ(k). ?M`(i) = N`(i)" using restrict_type2 restrict apply_funtype by auto with A5 `?M : succ(k) → FinPow(X)`have "?M {is partition}" using func1_1_L1 Partition_def by simp with A3 `?M : succ(k) → FinPow(X)` have "(∏ {⟨i,\<pr>(?M`(i),a)⟩. i ∈ succ(k)}) = (\<pr>(⋃i ∈ succ(k). ?M`(i),a))" by blast with III show ?thesis by simp qed also have "… = (\<pr>(⋃i ∈ succ(succ(k)). N`(i),a))" proof - let ?A = "⋃i ∈ succ(k). N`(i)" let ?B = "N`(succ(k))" from A4 `k ∈ nat` have "succ(k) ∈ nat" and "∀i ∈ succ(k). N`(i) ∈ FinPow(X)" using apply_funtype by auto then have "?A ∈ FinPow(X)" by (rule union_fin_list_fin) moreover from I have "?A ≠ 0" by auto moreover from A4 I have "N`(succ(k)) ∈ FinPow(X)" and "N`(succ(k)) ≠ 0" using apply_funtype by auto moreover from `succ(k) ∈ nat` A4 A5 have "?A ∩ ?B = 0" by (rule list_partition) moreover note A1 ultimately have "\<pr>(?A∪?B,a) = (\<pr>(?A,a))⋅(\<pr>(?B,a))" using prod_disjoint by simp moreover have "?A ∪ ?B = (⋃i ∈ succ(succ(k)). N`(i))" by auto ultimately show ?thesis by simp qed finally have "(∏ {⟨i,\<pr>(N`(i),a)⟩. i ∈ succ(succ(k))}) = (\<pr>(⋃i ∈ succ(succ(k)). N`(i),a))" by simp } thus ?thesis by auto qed } thus ?thesis by simp qed ultimately show ?thesis by (rule ind_on_nat) qed text{*A more convenient reformulation of @{text "prod_list_of_lists"}. *} theorem (in semigr1) prod_list_of_sets: assumes A1: "f {is commutative on} G" and A2: "n ∈ nat" "n ≠ 0" and A3: "M : n → FinPow(X)" "M {is partition}" shows "(∏ {⟨i,\<pr>(M`(i),a)⟩. i ∈ n}) = (\<pr>(⋃i ∈ n. M`(i),a))" proof - from A2 obtain k where "k ∈ nat" and "n = succ(k)" using Nat_ZF_1_L3 by auto with A1 A3 show ?thesis using prod_list_of_lists by simp qed text{*The definition of the product @{text "\<pr>(A,a) ≡ SetFold(f,a,A,r)"} of a some (finite) set of semigroup elements requires that $r$ is a linear order on the set of indices $A$. This is necessary so that we know in which order we are multiplying the elements. The product over $A$ is defined so that we have $\prod_A a = \prod a \circ \sigma(A)$ where $\sigma : |A| \rightarrow A$ is the enumeration of $A$ (the only order isomorphism between the number of elements in $A$ and $A$), see lemma @{text "setproddef"}. However, if the operation is commutative, the order is irrelevant. The next theorem formalizes that fact stating that we can replace the enumeration $\sigma (A)$ by any bijection between $|A|$ and $A$. In a way this is a generalization of @{text "setproddef"}. The proof is based on application of @{text "prod_list_of_sets"} to the finite collection of singletons that comprise $A$.*} theorem (in semigr1) prod_order_irr: assumes A1: "f {is commutative on} G" and A2: "A ∈ FinPow(X)" "A ≠ 0" and A3: "b ∈ bij(|A|,A)" shows "(∏ (a O b)) = \<pr>(A,a)" proof - let ?n = "|A|" let ?M = "{⟨k, {b`(k)}⟩. k ∈ ?n}" have "(∏ (a O b)) = (∏ {⟨i,\<pr>(?M`(i),a)⟩. i ∈ ?n})" proof - have "∀i ∈ ?n. \<pr>(?M`(i),a) = (a O b)`(i)" proof fix i assume "i ∈ ?n" with A2 A3 `i ∈ ?n` have "b`(i) ∈ X" using bij_def inj_def apply_funtype FinPow_def by auto then have "\<pr>({b`(i)},a) = a`(b`(i))" using gen_prod_singleton by simp with A3 `i ∈ ?n` have "\<pr>({b`(i)},a) = (a O b)`(i)" using bij_def inj_def comp_fun_apply by auto with `i ∈ ?n` A3 show "\<pr>(?M`(i),a) = (a O b)`(i)" using bij_def inj_partition by auto qed hence "{⟨i,\<pr>(?M`(i),a)⟩. i ∈ ?n} = {⟨i,(a O b)`(i)⟩. i ∈ ?n}" by simp moreover have "{⟨i,(a O b)`(i)⟩. i ∈ ?n} = a O b" proof - from A3 have "b : ?n → A" using bij_def inj_def by simp moreover from A2 have "A ⊆ X" using FinPow_def by simp ultimately have "b : ?n → X" by (rule func1_1_L1B) then have "a O b: ?n → G" using a_is_fun comp_fun by simp then show "{⟨i,(a O b)`(i)⟩. i ∈ ?n} = a O b" using fun_is_set_of_pairs by simp qed ultimately show ?thesis by simp qed also have "… = (\<pr>(⋃i ∈ ?n. ?M`(i),a))" proof - note A1 moreover from A2 have "?n ∈ nat" and "?n ≠ 0" using card_fin_is_nat card_non_empty_non_zero by auto moreover have "?M : ?n → FinPow(X)" and "?M {is partition}" proof - from A2 A3 have "∀k ∈ ?n. {b`(k)} ∈ FinPow(X)" using bij_def inj_def apply_funtype FinPow_def singleton_in_finpow by auto then show "?M : ?n → FinPow(X)" using ZF_fun_from_total by simp from A3 show "?M {is partition}" using bij_def inj_partition by auto qed ultimately show "(∏ {⟨i,\<pr>(?M`(i),a)⟩. i ∈ ?n}) = (\<pr>(⋃i ∈ ?n. ?M`(i),a))" by (rule prod_list_of_sets) qed also from A3 have "(\<pr>(⋃i ∈ ?n. ?M`(i),a)) = \<pr>(A,a)" using bij_def inj_partition surj_singleton_image by auto finally show ?thesis by simp qed text{*Another way of expressing the fact that the product dos not depend on the order. *} corollary (in semigr1) prod_bij_same: assumes "f {is commutative on} G" and "A ∈ FinPow(X)" "A ≠ 0" and "b ∈ bij(|A|,A)" "c ∈ bij(|A|,A)" shows "(∏ (a O b)) = (∏ (a O c))" using assms prod_order_irr by simp end