(* This file is a part of IsarMathLib - a library of formalized mathematics for Isabelle/Isar. Copyright (C) 2007 Slawomir Kolodynski This program 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. *) header{*\isaheader{FiniteSeq\_ZF.thy}*} theory FiniteSeq_ZF imports Nat_ZF_IML func1 begin text{* This theory treats finite sequences (i.e. maps $n\rightarrow X$, where $n=\{0,1,..,n-1\}$ is a natural number) as lists. It defines and proves the properties of basic operations on lists: concatenation, appending and element etc.*} section{*Lists as finite sequences*} text{*A natural way of representing (finite) lists in set theory is through (finite) sequences. In such view a list of elements of a set $X$ is a function that maps the set $\{0,1,..n-1\}$ into $X$. Since natural numbers in set theory are defined so that $n =\{0,1,..n-1\}$, a list of length $n$ can be understood as an element of the function space $n\rightarrow X$. *} text{*We define the set of lists with values in set $X$ as @{text "Lists(X)"}.*} definition "Lists(X) ≡ ⋃n∈nat.(n→X)" text{*The set of nonempty $X$-value listst will be called @{text "NELists(X)"}.*} definition "NELists(X) ≡ ⋃n∈nat.(succ(n)→X)" text{*We first define the shift that moves the second sequence to the domain $\{n,..,n+k-1\}$, where $n,k$ are the lengths of the first and the second sequence, resp. To understand the notation in the definitions below recall that in Isabelle/ZF @{text "pred(n)"} is the previous natural number and denotes the difference between natural numbers $n$ and $k$.*} definition "ShiftedSeq(b,n) ≡ {⟨j, b`(j #- n)⟩. j ∈ NatInterval(n,domain(b))}" text{*We define concatenation of two sequences as the union of the first sequence with the shifted second sequence. The result of concatenating lists $a$ and $b$ is called @{text "Concat(a,b)"}. *} definition "Concat(a,b) ≡ a ∪ ShiftedSeq(b,domain(a))" text{* For a finite sequence we define the sequence of all elements except the first one. This corresponds to the "tail" function in Haskell. We call it @{text "Tail"} here as well.*} definition "Tail(a) ≡ {⟨k, a`(succ(k))⟩. k ∈ pred(domain(a))}" text{*A dual notion to @{text "Tail"} is the list of all elements of a list except the last one. Borrowing the terminology from Haskell again, we will call this @{text "Init"}.*} definition "Init(a) ≡ restrict(a,pred(domain(a)))" text{* Another obvious operation we can talk about is appending an element at the end of a sequence. This is called @{text "Append"}.*} definition "Append(a,x) ≡ a ∪ {⟨domain(a),x⟩}" text{*If lists are modeled as finite sequences (i.e. functions on natural intervals $\{0,1,..,n-1\} = n$) it is easy to get the first element of a list as the value of the sequence at $0$. The last element is the value at $n-1$. To hide this behind a familiar name we define the @{text "Last"} element of a list. *} definition "Last(a) ≡ a`(pred(domain(a)))" text{*Shifted sequence is a function on a the interval of natural numbers.*} lemma shifted_seq_props: assumes A1: "n ∈ nat" "k ∈ nat" and A2: "b:k→X" shows "ShiftedSeq(b,n): NatInterval(n,k) → X" "∀i ∈ NatInterval(n,k). ShiftedSeq(b,n)`(i) = b`(i #- n)" "∀j∈k. ShiftedSeq(b,n)`(n #+ j) = b`(j)" proof - let ?I = "NatInterval(n,domain(b))" from A2 have Fact: "?I = NatInterval(n,k)" using func1_1_L1 by simp with A1 A2 have "∀j∈ ?I. b`(j #- n) ∈ X" using inter_diff_in_len apply_funtype by simp then have "{⟨j, b`(j #- n)⟩. j ∈ ?I} : ?I → X" by (rule ZF_fun_from_total) with Fact show thesis_1: "ShiftedSeq(b,n): NatInterval(n,k) → X" using ShiftedSeq_def by simp { fix i from Fact thesis_1 have "ShiftedSeq(b,n): ?I → X" by simp moreover assume "i ∈ NatInterval(n,k)" with Fact have "i ∈ ?I" by simp moreover from Fact have "ShiftedSeq(b,n) = {⟨i, b`(i #- n)⟩. i ∈ ?I}" using ShiftedSeq_def by simp ultimately have "ShiftedSeq(b,n)`(i) = b`(i #- n)" by (rule ZF_fun_from_tot_val) } then show thesis1: "∀i ∈ NatInterval(n,k). ShiftedSeq(b,n)`(i) = b`(i #- n)" by simp { fix j let ?i = "n #+ j" assume A3: "j∈k" with A1 have "j ∈ nat" using elem_nat_is_nat by blast then have "?i #- n = j" using diff_add_inverse by simp with A3 thesis1 have "ShiftedSeq(b,n)`(?i) = b`(j)" using NatInterval_def by auto } then show "∀j∈k. ShiftedSeq(b,n)`(n #+ j) = b`(j)" by simp qed text{*Basis properties of the contatenation of two finite sequences.*} theorem concat_props: assumes A1: "n ∈ nat" "k ∈ nat" and A2: "a:n→X" "b:k→X" shows "Concat(a,b): n #+ k → X" "∀i∈n. Concat(a,b)`(i) = a`(i)" "∀i ∈ NatInterval(n,k). Concat(a,b)`(i) = b`(i #- n)" "∀j ∈ k. Concat(a,b)`(n #+ j) = b`(j)" proof - from A1 A2 have "a:n→X" and I: "ShiftedSeq(b,n): NatInterval(n,k) → X" and "n ∩ NatInterval(n,k) = 0" using shifted_seq_props length_start_decomp by auto then have "a ∪ ShiftedSeq(b,n): n ∪ NatInterval(n,k) → X ∪ X" by (rule fun_disjoint_Un) with A1 A2 show "Concat(a,b): n #+ k → X" using func1_1_L1 Concat_def length_start_decomp by auto { fix i assume "i ∈ n" with A1 I have "i ∉ domain(ShiftedSeq(b,n))" using length_start_decomp func1_1_L1 by auto with A2 have "Concat(a,b)`(i) = a`(i)" using func1_1_L1 fun_disjoint_apply1 Concat_def by simp } thus "∀i∈n. Concat(a,b)`(i) = a`(i)" by simp { fix i assume A3: "i ∈ NatInterval(n,k)" with A1 A2 have "i ∉ domain(a)" using length_start_decomp func1_1_L1 by auto with A1 A2 A3 have "Concat(a,b)`(i) = b`(i #- n)" using func1_1_L1 fun_disjoint_apply2 Concat_def shifted_seq_props by simp } thus II: "∀i ∈ NatInterval(n,k). Concat(a,b)`(i) = b`(i #- n)" by simp { fix j let ?i = "n #+ j" assume A3: "j∈k" with A1 have "j ∈ nat" using elem_nat_is_nat by blast then have "?i #- n = j" using diff_add_inverse by simp with A3 II have "Concat(a,b)`(?i) = b`(j)" using NatInterval_def by auto } thus "∀j ∈ k. Concat(a,b)`(n #+ j) = b`(j)" by simp qed text{*Properties of concatenating three lists.*} lemma concat_concat_list: assumes A1: "n ∈ nat" "k ∈ nat" "m ∈ nat" and A2: "a:n→X" "b:k→X" "c:m→X" and A3: "d = Concat(Concat(a,b),c)" shows "d : n #+k #+ m → X" "∀j ∈ n. d`(j) = a`(j)" "∀j ∈ k. d`(n #+ j) = b`(j)" "∀j ∈ m. d`(n #+ k #+ j) = c`(j)" proof - from A1 A2 have I: "n #+ k ∈ nat" "m ∈ nat" "Concat(a,b): n #+ k → X" "c:m→X" using concat_props by auto with A3 show "d: n #+k #+ m → X" using concat_props by simp from I have II: "∀i ∈ n #+ k. Concat(Concat(a,b),c)`(i) = Concat(a,b)`(i)" by (rule concat_props) { fix j assume A4: "j ∈ n" moreover from A1 have "n ⊆ n #+ k" using add_nat_le by simp ultimately have "j ∈ n #+ k" by auto with A3 II have "d`(j) = Concat(a,b)`(j)" by simp with A1 A2 A4 have "d`(j) = a`(j)" using concat_props by simp } thus "∀j ∈ n. d`(j) = a`(j)" by simp { fix j assume A5: "j ∈ k" with A1 A3 II have "d`(n #+ j) = Concat(a,b)`(n #+ j)" using add_lt_mono by simp also from A1 A2 A5 have "… = b`(j)" using concat_props by simp finally have "d`(n #+ j) = b`(j)" by simp } thus "∀j ∈ k. d`(n #+ j) = b`(j)" by simp from I have "∀j ∈ m. Concat(Concat(a,b),c)`(n #+ k #+ j) = c`(j)" by (rule concat_props) with A3 show "∀j ∈ m. d`(n #+ k #+ j) = c`(j)" by simp qed text{*Properties of concatenating a list with a concatenation of two other lists.*} lemma concat_list_concat: assumes A1: "n ∈ nat" "k ∈ nat" "m ∈ nat" and A2: "a:n→X" "b:k→X" "c:m→X" and A3: "e = Concat(a, Concat(b,c))" shows "e : n #+k #+ m → X" "∀j ∈ n. e`(j) = a`(j)" "∀j ∈ k. e`(n #+ j) = b`(j)" "∀j ∈ m. e`(n #+ k #+ j) = c`(j)" proof - from A1 A2 have I: "n ∈ nat" "k #+ m ∈ nat" "a:n→X" "Concat(b,c): k #+ m → X" using concat_props by auto with A3 show "e : n #+k #+ m → X" using concat_props add_assoc by simp from I have "∀j ∈ n. Concat(a, Concat(b,c))`(j) = a`(j)" by (rule concat_props) with A3 show "∀j ∈ n. e`(j) = a`(j)" by simp from I have II: "∀j ∈ k #+ m. Concat(a, Concat(b,c))`(n #+ j) = Concat(b,c)`(j)" by (rule concat_props) { fix j assume A4: "j ∈ k" moreover from A1 have "k ⊆ k #+ m" using add_nat_le by simp ultimately have "j ∈ k #+ m" by auto with A3 II have "e`(n #+ j) = Concat(b,c)`(j)" by simp also from A1 A2 A4 have "… = b`(j)" using concat_props by simp finally have "e`(n #+ j) = b`(j)" by simp } thus "∀j ∈ k. e`(n #+ j) = b`(j)" by simp { fix j assume A5: "j ∈ m" with A1 II A3 have "e`(n #+ k #+ j) = Concat(b,c)`(k #+ j)" using add_lt_mono add_assoc by simp also from A1 A2 A5 have "… = c`(j)" using concat_props by simp finally have "e`(n #+ k #+ j) = c`(j)" by simp } then show "∀j ∈ m. e`(n #+ k #+ j) = c`(j)" by simp qed text{*Concatenation is associative.*} theorem concat_assoc: assumes A1: "n ∈ nat" "k ∈ nat" "m ∈ nat" and A2: "a:n→X" "b:k→X" "c:m→X" shows "Concat(Concat(a,b),c) = Concat(a, Concat(b,c))" proof - let ?d = "Concat(Concat(a,b),c)" let ?e = "Concat(a, Concat(b,c))" from A1 A2 have "?d : n #+k #+ m → X" and "?e : n #+k #+ m → X" using concat_concat_list concat_list_concat by auto moreover have "∀i ∈ n #+k #+ m. ?d`(i) = ?e`(i)" proof - { fix i assume "i ∈ n #+k #+ m" moreover from A1 have "n #+k #+ m = n ∪ NatInterval(n,k) ∪ NatInterval(n #+ k,m)" using adjacent_intervals3 by simp ultimately have "i ∈ n ∨ i ∈ NatInterval(n,k) ∨ i ∈ NatInterval(n #+ k,m)" by simp moreover { assume "i ∈ n" with A1 A2 have "?d`(i) = ?e`(i)" using concat_concat_list concat_list_concat by simp } moreover { assume "i ∈ NatInterval(n,k)" then obtain j where "j∈k" and "i = n #+ j" using NatInterval_def by auto with A1 A2 have "?d`(i) = ?e`(i)" using concat_concat_list concat_list_concat by simp } moreover { assume "i ∈ NatInterval(n #+ k,m)" then obtain j where "j ∈ m" and "i = n #+ k #+ j" using NatInterval_def by auto with A1 A2 have "?d`(i) = ?e`(i)" using concat_concat_list concat_list_concat by simp } ultimately have "?d`(i) = ?e`(i)" by auto } thus ?thesis by simp qed ultimately show "?d = ?e" by (rule func_eq) qed text{*Properties of @{text "Tail"}.*} theorem tail_props: assumes A1: "n ∈ nat" and A2: "a: succ(n) → X" shows "Tail(a) : n → X" "∀k ∈ n. Tail(a)`(k) = a`(succ(k))" proof - from A1 A2 have "∀k ∈ n. a`(succ(k)) ∈ X" using succ_ineq apply_funtype by simp then have "{⟨k, a`(succ(k))⟩. k ∈ n} : n → X" by (rule ZF_fun_from_total) with A2 show I: "Tail(a) : n → X" using func1_1_L1 pred_succ_eq Tail_def by simp moreover from A2 have "Tail(a) = {⟨k, a`(succ(k))⟩. k ∈ n}" using func1_1_L1 pred_succ_eq Tail_def by simp ultimately show "∀k ∈ n. Tail(a)`(k) = a`(succ(k))" by (rule ZF_fun_from_tot_val0) qed text{*Properties of @{text "Append"}. It is a bit surprising that the we don't need to assume that $n$ is a natural number.*} theorem append_props: assumes A1: "a: n → X" and A2: "x∈X" and A3: "b = Append(a,x)" shows "b : succ(n) → X" "∀k∈n. b`(k) = a`(k)" "b`(n) = x" proof - note A1 moreover have I: "n ∉ n" using mem_not_refl by simp moreover from A1 A3 have II: "b = a ∪ {⟨n,x⟩}" using func1_1_L1 Append_def by simp ultimately have "b : n ∪ {n} → X ∪ {x}" by (rule func1_1_L11D) with A2 show "b : succ(n) → X" using succ_explained set_elem_add by simp from A1 I II show "∀k∈n. b`(k) = a`(k)" and "b`(n) = x" using func1_1_L11D by auto qed text{*A special case of @{text "append_props"}: appending to a nonempty list does not change the head (first element) of the list.*} corollary head_of_append: assumes "n∈ nat" and "a: succ(n) → X" and "x∈X" shows "Append(a,x)`(0) = a`(0)" using assms append_props empty_in_every_succ by auto (*text{*A bit technical special case of @{text "append_props"} that tells us what is the value of the appended list at the sucessor of some argument.*} corollary append_val_succ: assumes "n ∈ nat" and "a: succ(n) → X" and "x∈X" and "k ∈ n" shows "Append(a,x)`(succ(k)) = a`(succ(k))" using assms succ_ineq append_props by simp*) text{* @{text "Tail"} commutes with @{text "Append"}.*} theorem tail_append_commute: assumes A1: "n ∈ nat" and A2: "a: succ(n) → X" and A3: "x∈X" shows "Append(Tail(a),x) = Tail(Append(a,x))" proof - let ?b = "Append(Tail(a),x)" let ?c = "Tail(Append(a,x))" from A1 A2 have I: "Tail(a) : n → X" using tail_props by simp from A1 A2 A3 have "succ(n) ∈ nat" and "Append(a,x) : succ(succ(n)) → X" using append_props by auto then have II: "∀k ∈ succ(n). ?c`(k) = Append(a,x)`(succ(k))" by (rule tail_props) from assms have "?b : succ(n) → X" and "?c : succ(n) → X" using tail_props append_props by auto moreover have "∀k ∈ succ(n). ?b`(k) = ?c`(k)" proof - { fix k assume "k ∈ succ(n)" hence "k ∈ n ∨ k = n" by auto moreover { assume A4: "k ∈ n" with assms II have "?c`(k) = a`(succ(k))" using succ_ineq append_props by simp moreover from A3 I have "∀k∈n. ?b`(k) = Tail(a)`(k)" using append_props by simp with A1 A2 A4 have "?b`(k) = a`(succ(k))" using tail_props by simp ultimately have "?b`(k) = ?c`(k)" by simp } moreover { assume A5: "k = n" with A2 A3 I II have "?b`(k) = ?c`(k)" using append_props by auto } ultimately have "?b`(k) = ?c`(k)" by auto } thus ?thesis by simp qed ultimately show "?b = ?c" by (rule func_eq) qed text{*Properties of @{text "Init"}.*} theorem init_props: assumes A1: "n ∈ nat" and A2: "a: succ(n) → X" shows "Init(a) : n → X" "∀k∈n. Init(a)`(k) = a`(k)" "a = Append(Init(a), a`(n))" proof - have "n ⊆ succ(n)" by auto with A2 have "restrict(a,n): n → X" using restrict_type2 by simp moreover from A1 A2 have I: "restrict(a,n) = Init(a)" using func1_1_L1 pred_succ_eq Init_def by simp ultimately show thesis1: "Init(a) : n → X" by simp { fix k assume "k∈n" then have "restrict(a,n)`(k) = a`(k)" using restrict by simp with I have "Init(a)`(k) = a`(k)" by simp } then show thesis2: "∀k∈n. Init(a)`(k) = a`(k)" by simp let ?b = "Append(Init(a), a`(n))" from A2 thesis1 have II: "Init(a) : n → X" "a`(n) ∈ X" "?b = Append(Init(a), a`(n))" using apply_funtype by auto note A2 moreover from II have "?b : succ(n) → X" by (rule append_props) moreover have "∀k ∈ succ(n). a`(k) = ?b`(k)" proof - { fix k assume A3: "k ∈ n" from II have "∀j∈n. ?b`(j) = Init(a)`(j)" by (rule append_props) with thesis2 A3 have "a`(k) = ?b`(k)" by simp } moreover from II have "?b`(n) = a`(n)" by (rule append_props) hence " a`(n) = ?b`(n)" by simp ultimately show "∀k ∈ succ(n). a`(k) = ?b`(k)" by simp qed ultimately show "a = ?b" by (rule func_eq) qed text{*If we take init of the result of append, we get back the same list.*} lemma init_append: assumes A1: "n ∈ nat" and A2: "a:n→X" and A3: "x ∈ X" shows "Init(Append(a,x)) = a" proof - from A2 A3 have "Append(a,x): succ(n)→X" using append_props by simp with A1 have "Init(Append(a,x)):n→X" and "∀k∈n. Init(Append(a,x))`(k) = Append(a,x)`(k)" using init_props by auto with A2 A3 have "∀k∈n. Init(Append(a,x))`(k) = a`(k)" using append_props by simp with `Init(Append(a,x)):n→X` A2 show ?thesis by (rule func_eq) qed text{*A reformulation of definition of @{text "Init"}.*} lemma init_def: assumes "n ∈ nat" and "x:succ(n)→X" shows "Init(x) = restrict(x,n)" using assms func1_1_L1 Init_def by simp text{*A lemma about extending a finite sequence by one more value. This is just a more explicit version of @{text "append_props"}.*} lemma finseq_extend: assumes "a:n→X" "y∈X" "b = a ∪ {⟨n,y⟩}" shows "b: succ(n) → X" "∀k∈n. b`(k) = a`(k)" "b`(n) = y" using assms Append_def func1_1_L1 append_props by auto text{*The next lemma is a bit displaced as it is mainly about finite sets. It is proven here because it uses the notion of @{text "Append"}. Suppose we have a list of element of $A$ is a bijection. Then for every element that does not belong to $A$ we can we can construct a bijection for the set $A \cup \{ x\}$ by appending $x$. This is just a specialised version of lemma @{text "bij_extend_point"} from @{text "func1.thy"}. *} lemma bij_append_point: assumes A1: "n ∈ nat" and A2: "b ∈ bij(n,X)" and A3: "x ∉ X" shows "Append(b,x) ∈ bij(succ(n), X ∪ {x})" proof - from A2 A3 have "b ∪ {⟨n,x⟩} ∈ bij(n ∪ {n},X ∪ {x})" using mem_not_refl bij_extend_point by simp moreover have "Append(b,x) = b ∪ {⟨n,x⟩}" proof - from A2 have "b:n→X" using bij_def surj_def by simp then have "b : n → X ∪ {x}" using func1_1_L1B by blast then show "Append(b,x) = b ∪ {⟨n,x⟩}" using Append_def func1_1_L1 by simp qed ultimately show ?thesis using succ_explained by auto qed text{*The next lemma rephrases the definition of @{text "Last"}. Recall that in ZF we have $\{0,1,2,..,n\} = n+1=$@{text "succ"}$(n)$.*} lemma last_seq_elem: assumes "a: succ(n) → X" shows "Last(a) = a`(n)" using assms func1_1_L1 pred_succ_eq Last_def by simp text{*If two finite sequences are the same when restricted to domain one shorter than the original and have the same value on the last element, then they are equal.*} lemma finseq_restr_eq: assumes A1: "n ∈ nat" and A2: "a: succ(n) → X" "b: succ(n) → X" and A3: "restrict(a,n) = restrict(b,n)" and A4: "a`(n) = b`(n)" shows "a = b" proof - { fix k assume "k ∈ succ(n)" then have "k ∈ n ∨ k = n" by auto moreover { assume "k ∈ n" then have "restrict(a,n)`(k) = a`(k)" and "restrict(b,n)`(k) = b`(k)" using restrict by auto with A3 have "a`(k) = b`(k)" by simp } moreover { assume "k = n" with A4 have "a`(k) = b`(k)" by simp } ultimately have "a`(k) = b`(k)" by auto } then have "∀ k ∈ succ(n). a`(k) = b`(k)" by simp with A2 show "a = b" by (rule func_eq) qed text{*Concatenating a list of length $1$ is the same as appending its first (and only) element. Recall that in ZF set theory $1 = \{ 0 \} $.*} lemma append_1elem: assumes A1: "n ∈ nat" and A2: "a: n → X" and A3: "b : 1 → X" shows "Concat(a,b) = Append(a,b`(0))" proof - let ?C = "Concat(a,b)" let ?A = "Append(a,b`(0))" from A1 A2 A3 have I: "n ∈ nat" "1 ∈ nat" "a:n→X" "b:1→X" by auto have "?C : succ(n) → X" proof - from I have "?C : n #+ 1 → X" by (rule concat_props) with A1 show "?C : succ(n) → X" by simp qed moreover from A2 A3 have "?A : succ(n) → X" using apply_funtype append_props by simp moreover have "∀k ∈ succ(n). ?C`(k) = ?A`(k)" proof fix k assume "k ∈ succ(n)" moreover { assume "k ∈ n" moreover from I have "∀i ∈ n. ?C`(i) = a`(i)" by (rule concat_props) moreover from A2 A3 have "∀i∈n. ?A`(i) = a`(i)" using apply_funtype append_props by simp ultimately have "?C`(k) = ?A`(k)" by simp } moreover have "?C`(n) = ?A`(n)" proof - from I have "∀j ∈ 1. ?C`(n #+ j) = b`(j)" by (rule concat_props) with A1 A2 A3 show "?C`(n) = ?A`(n)" using apply_funtype append_props by simp qed ultimately show "?C`(k) = ?A`(k)" by auto qed ultimately show "?C = ?A" by (rule func_eq) qed text{*A simple lemma about lists of length $1$.*} lemma list_len1_singleton: assumes A1: "x∈X" shows "{⟨0,x⟩} : 1 → X" proof - from A1 have "{⟨0,x⟩} : {0} → X" using pair_func_singleton by simp moreover have "{0} = 1" by auto ultimately show ?thesis by simp qed text{*A singleton list is in fact a singleton set with a pair as the only element.*} lemma list_singleton_pair: assumes A1: "x:1→X" shows "x = {⟨0,x`(0)⟩}" proof - from A1 have "x = {⟨t,x`(t)⟩. t∈1}" by (rule fun_is_set_of_pairs) hence "x = {⟨t,x`(t)⟩. t∈{0} }" by simp thus ?thesis by simp qed text{*When we append an element to the empty list we get a list with length $1$.*} lemma empty_append1: assumes A1: "x∈X" shows "Append(0,x): 1 → X" and "Append(0,x)`(0) = x" proof - let ?a = "Append(0,x)" have "?a = {⟨0,x⟩}" using Append_def by auto with A1 show "?a : 1 → X" and "?a`(0) = x" using list_len1_singleton pair_func_singleton by auto qed (*text{*Tail of a list of length 1 is a list of length 0.*} lemma list_len1_tail: assumes "a:1→X" shows "Tail(a) : 0 → X" using assms tail_props by blast *) text{*Appending an element is the same as concatenating with certain pair.*} lemma append_concat_pair: assumes "n ∈ nat" and "a: n → X" and "x∈X" shows "Append(a,x) = Concat(a,{⟨0,x⟩})" using assms list_len1_singleton append_1elem pair_val by simp text{*An associativity property involving concatenation and appending. For proof we just convert appending to concatenation and use @{text "concat_assoc"}.*} lemma concat_append_assoc: assumes A1: "n ∈ nat" "k ∈ nat" and A2: "a:n→X" "b:k→X" and A3: "x ∈ X" shows "Append(Concat(a,b),x) = Concat(a, Append(b,x))" proof - from A1 A2 A3 have "n #+ k ∈ nat" "Concat(a,b) : n #+ k → X" "x ∈ X" using concat_props by auto then have "Append(Concat(a,b),x) = Concat(Concat(a,b),{⟨0,x⟩})" by (rule append_concat_pair) moreover from A1 A2 A3 have "n ∈ nat" "k ∈ nat" "1 ∈ nat" "a:n→X" "b:k→X" "{⟨0,x⟩} : 1 → X" using list_len1_singleton by auto then have "Concat(Concat(a,b),{⟨0,x⟩}) = Concat(a, Concat(b,{⟨0,x⟩}))" by (rule concat_assoc) moreover from A1 A2 A3 have "Concat(b,{⟨0,x⟩}) = Append(b,x)" using list_len1_singleton append_1elem pair_val by simp ultimately show "Append(Concat(a,b),x) = Concat(a, Append(b,x))" by simp qed text{*An identity involving concatenating with init and appending the last element.*} lemma concat_init_last_elem: assumes "n ∈ nat" "k ∈ nat" and "a: n → X" and "b : succ(k) → X" shows "Append(Concat(a,Init(b)),b`(k)) = Concat(a,b)" using assms init_props apply_funtype concat_append_assoc by simp text{*A lemma about creating lists by composition and how @{text "Append"} behaves in such case.*} lemma list_compose_append: assumes A1: "n ∈ nat" and A2: "a : n → X" and A3: "x ∈ X" and A4: "c : X → Y" shows "c O Append(a,x) : succ(n) → Y" "c O Append(a,x) = Append(c O a, c`(x))" proof - let ?b = "Append(a,x)" let ?d = "Append(c O a, c`(x))" from A2 A4 have "c O a : n → Y" using comp_fun by simp from A2 A3 have "?b : succ(n) → X" using append_props by simp with A4 show "c O ?b : succ(n) → Y" using comp_fun by simp moreover from A3 A4 `c O a : n → Y` have "?d: succ(n) → Y" using apply_funtype append_props by simp moreover have "∀k ∈ succ(n). (c O ?b) `(k) = ?d`(k)" proof - { fix k assume "k ∈ succ(n)" with `?b : succ(n) → X` have "(c O ?b) `(k) = c`(?b`(k))" using comp_fun_apply by simp with A2 A3 A4 `c O a : n → Y` `c O a : n → Y` `k ∈ succ(n)` have "(c O ?b) `(k) = ?d`(k)" using append_props comp_fun_apply apply_funtype by auto } thus ?thesis by simp qed ultimately show "c O ?b = ?d" by (rule func_eq) qed text{*A lemma about appending an element to a list defined by set comprehension.*} lemma set_list_append: assumes A1: "∀i ∈ succ(k). b(i) ∈ X" and A2: "a = {⟨i,b(i)⟩. i ∈ succ(k)}" shows "a: succ(k) → X" "{⟨i,b(i)⟩. i ∈ k}: k → X" "a = Append({⟨i,b(i)⟩. i ∈ k},b(k))" proof - from A1 have "{⟨i,b(i)⟩. i ∈ succ(k)} : succ(k) → X" by (rule ZF_fun_from_total) with A2 show "a: succ(k) → X" by simp from A1 have "∀i ∈ k. b(i) ∈ X" by simp then show "{⟨i,b(i)⟩. i ∈ k}: k → X" by (rule ZF_fun_from_total) with A2 show "a = Append({⟨i,b(i)⟩. i ∈ k},b(k))" using func1_1_L1 Append_def by auto qed text{* An induction theorem for lists.*} lemma list_induct: assumes A1: "∀b∈1→X. P(b)" and A2: "∀b∈NELists(X). P(b) ⟶ (∀x∈X. P(Append(b,x)))" and A3: "d ∈ NELists(X)" shows "P(d)" proof - { fix n assume "n∈nat" moreover from A1 have "∀b∈succ(0)→X. P(b)" by simp moreover have "∀k∈nat. ((∀b∈succ(k)→X. P(b)) ⟶ (∀c∈succ(succ(k))→X. P(c)))" proof - { fix k assume "k ∈ nat" assume "∀b∈succ(k)→X. P(b)" have "∀c∈succ(succ(k))→X. P(c)" proof fix c assume "c: succ(succ(k))→X" let ?b = "Init(c)" let ?x = "c`(succ(k))" from `k ∈ nat` `c: succ(succ(k))→X` have "?b:succ(k)→X" using init_props by simp with A2 `k ∈ nat` `∀b∈succ(k)→X. P(b)` have "∀x∈X. P(Append(?b,x))" using NELists_def by auto with `c: succ(succ(k))→X` have "P(Append(?b,?x))" using apply_funtype by simp with `k ∈ nat` `c: succ(succ(k))→X` show "P(c)" using init_props by simp qed } thus ?thesis by simp qed ultimately have "∀b∈succ(n)→X. P(b)" by (rule ind_on_nat) } with A3 show ?thesis using NELists_def by auto qed section{*Lists and cartesian products*} text{*Lists of length $n$ of elements of some set $X$ can be thought of as a model of the cartesian product $X^n$ which is more convenient in many applications.*} text{*There is a natural bijection between the space $(n+1)\rightarrow X$ of lists of length $n+1$ of elements of $X$ and the cartesian product $(n\rightarrow X)\times X$.*} lemma lists_cart_prod: assumes "n ∈ nat" shows "{⟨x,⟨Init(x),x`(n)⟩⟩. x ∈ succ(n)→X} ∈ bij(succ(n)→X,(n→X)×X)" proof - let ?f = "{⟨x,⟨Init(x),x`(n)⟩⟩. x ∈ succ(n)→X}" from assms have "∀x ∈ succ(n)→X. ⟨Init(x),x`(n)⟩ ∈ (n→X)×X" using init_props succ_iff apply_funtype by simp then have I: "?f: (succ(n)→X)→((n→X)×X)" by (rule ZF_fun_from_total) moreover from assms I have "∀x∈succ(n)→X.∀y∈succ(n)→X. ?f`(x)=?f`(y) ⟶ x=y" using ZF_fun_from_tot_val init_def finseq_restr_eq by auto moreover have "∀p∈(n→X)×X.∃x∈succ(n)→X. ?f`(x) = p" proof fix p assume "p ∈ (n→X)×X" let ?x = "Append(fst(p),snd(p))" from assms `p ∈ (n→X)×X` have "?x:succ(n)→X" using append_props by simp with I have "?f`(?x) = ⟨Init(?x),?x`(n)⟩" using succ_iff ZF_fun_from_tot_val by simp moreover from assms `p ∈ (n→X)×X` have "Init(?x) = fst(p)" and "?x`(n) = snd(p)" using init_append append_props by auto ultimately have "?f`(?x) = ⟨fst(p),snd(p)⟩" by auto with `p ∈ (n→X)×X` `?x:succ(n)→X` show "∃x∈succ(n)→X. ?f`(x) = p" by auto qed ultimately show ?thesis using inj_def surj_def bij_def by auto qed text{*We can identify a set $X$ with lists of length one of elements of $X$.*} lemma singleton_list_bij: shows "{⟨x,x`(0)⟩. x∈1→X} ∈ bij(1→X,X)" proof - let ?f = "{⟨x,x`(0)⟩. x∈1→X}" have "∀x∈1→X. x`(0) ∈ X" using apply_funtype by simp then have I: "?f:(1→X)→X" by (rule ZF_fun_from_total) moreover have "∀x∈1→X.∀y∈1→X. ?f`(x) = ?f`(y) ⟶ x=y" proof - { fix x y assume "x:1→X" "y:1→X" and "?f`(x) = ?f`(y)" with I have "x`(0) = y`(0)" using ZF_fun_from_tot_val by auto moreover from `x:1→X` `y:1→X` have "x = {⟨0,x`(0)⟩}" and "y = {⟨0,y`(0)⟩}" using list_singleton_pair by auto ultimately have "x=y" by simp } thus ?thesis by auto qed moreover have "∀y∈X. ∃x∈1→X. ?f`(x)=y" proof fix y assume "y∈X" let ?x = "{⟨0,y⟩}" from I `y∈X` have "?x:1→X" and "?f`(?x) = y" using list_len1_singleton ZF_fun_from_tot_val pair_val by auto thus "∃x∈1→X. ?f`(x)=y" by auto qed ultimately show ?thesis using inj_def surj_def bij_def by simp qed text{*We can identify a set of $X$-valued lists of length with $X$.*} lemma list_singleton_bij: shows "{⟨x,{⟨0,x⟩}⟩.x∈X} ∈ bij(X,1→X)" and "{⟨y,y`(0)⟩. y∈1→X} = converse({⟨x,{⟨0,x⟩}⟩.x∈X})" and "{⟨x,{⟨0,x⟩}⟩.x∈X} = converse({⟨y,y`(0)⟩. y∈1→X})" proof - let ?f = "{⟨y,y`(0)⟩. y∈1→X}" let ?g = "{⟨x,{⟨0,x⟩}⟩.x∈X}" have "1 = {0}" by auto then have "?f ∈ bij(1→X,X)" and "?g:X→(1→X)" using singleton_list_bij pair_func_singleton ZF_fun_from_total by auto moreover have "∀y∈1→X.?g`(?f`(y)) = y" proof fix y assume "y:1→X" have "?f:(1→X)→X" using singleton_list_bij bij_def inj_def by simp with `1 = {0}` `y:1→X` `?g:X→(1→X)` show "?g`(?f`(y)) = y" using ZF_fun_from_tot_val apply_funtype func_singleton_pair by simp qed ultimately show "?g ∈ bij(X,1→X)" and "?f = converse(?g)" and "?g = converse(?f)" using comp_conv_id by auto qed text{*What is the inverse image of a set by the natural bijection between $X$-valued singleton lists and $X$? *} lemma singleton_vimage: assumes "U⊆X" shows "{x∈1→X. x`(0) ∈ U} = { {⟨0,y⟩}. y∈U}" proof have "1 = {0}" by auto { fix x assume "x ∈ {x∈1→X. x`(0) ∈ U}" with `1 = {0}` have "x = {⟨0, x`(0)⟩}" using func_singleton_pair by auto } thus "{x∈1→X. x`(0) ∈ U} ⊆ { {⟨0,y⟩}. y∈U}" by auto { fix x assume "x ∈ { {⟨0,y⟩}. y∈U}" then obtain y where "x = {⟨0,y⟩}" and "y∈U" by auto with `1 = {0}` assms have "x:1→X" using pair_func_singleton by auto } thus "{ {⟨0,y⟩}. y∈U} ⊆ {x∈1→X. x`(0) ∈ U}" by auto qed text{*A technical lemma about extending a list by values from a set.*} lemma list_append_from: assumes A1: "n ∈ nat" and A2: "U ⊆ n→X" and A3: "V ⊆ X" shows "{x ∈ succ(n)→X. Init(x) ∈ U ∧ x`(n) ∈ V} = (⋃y∈V.{Append(x,y).x∈U})" proof - { fix x assume "x ∈ {x ∈ succ(n)→X. Init(x) ∈ U ∧ x`(n) ∈ V}" then have "x ∈ succ(n)→X" and "Init(x) ∈ U" and I: "x`(n) ∈ V" by auto let ?y = "x`(n)" from A1 and `x ∈ succ(n)→X` have "x = Append(Init(x),?y)" using init_props by simp with I and `Init(x) ∈ U` have "x ∈ (⋃y∈V.{Append(a,y).a∈U})" by auto } moreover { fix x assume "x ∈ (⋃y∈V.{Append(a,y).a∈U})" then obtain a y where "y∈V" and "a∈U" and "x = Append(a,y)" by auto with A2 A3 have "x: succ(n)→X" using append_props by blast from A2 A3 `y∈V` `a∈U` have "a:n→X" and "y∈X" by auto with A1 `a∈U` `y∈V` `x = Append(a,y)` have "Init(x) ∈ U" and "x`(n) ∈ V" using append_props init_append by auto with `x: succ(n)→X` have "x ∈ {x ∈ succ(n)→X. Init(x) ∈ U ∧ x`(n) ∈ V}" by auto } ultimately show ?thesis by blast qed end