Theory InductiveSeq_ZF

theory InductiveSeq_ZF
imports FiniteSeq_ZF
(* 
This file is a part of IsarMathLib -
a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2007 Slawomir Kolodynski

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header{*\isaheader{InductiveSeq\_ZF.thy}*}

theory InductiveSeq_ZF imports Nat_ZF_IML FiniteSeq_ZF

begin

text{*In this theory we discuss sequences defined by conditions of the form
$a_0 = x,\ a_{n+1} = f(a_n)$ and similar.*}


section{*Sequences defined by induction*}

text{*One way of defining a sequence (that is a function $a:\mathbb{N}\rightarrow X$)
is to provide the first element of the sequence and a function to find the next
value when we have the current one. This is usually called "defining a sequence
by induction". In this section we set up the notion of
a sequence defined by induction and prove the theorems needed to use it.*}


text{*First we define a helper notion of the sequence defined inductively up to a
given natural number $n$. *}


definition
"InductiveSequenceN(x,f,n) ≡
THE a. a: succ(n) -> domain(f) ∧ a`(0) = x ∧ (∀k∈n. a`(succ(k)) = f`(a`(k)))"


text{*From that we define the inductive sequence on the
whole set of natural numbers. Recall that in Isabelle/ZF the set of natural numbers
is denoted @{text "nat"}.*}


definition
"InductiveSequence(x,f) ≡ \<Union>n∈nat. InductiveSequenceN(x,f,n)";

text{*First we will consider the question of existence and uniqueness
of finite inductive sequences. The proof
is by induction and the next lemma is the $P(0)$ step. To understand the notation
recall that for natural numbers in set theory we have $n = \{0,1,..,n-1\}$ and
@{text "succ(n)"}$ = \{0,1,..,n\}$.*}


lemma indseq_exun0: assumes A1: "f: X->X" and A2: "x∈X"
shows
"∃! a. a: succ(0) -> X ∧ a`(0) = x ∧ ( ∀k∈0. a`(succ(k)) = f`(a`(k)) )"
proof
fix a b
assume A3:
"a: succ(0) -> X ∧ a`(0) = x ∧ ( ∀k∈0. a`(succ(k)) = f`(a`(k)) )"
"b: succ(0) -> X ∧ b`(0) = x ∧ ( ∀k∈0. b`(succ(k)) = f`(b`(k)) )"
moreover have "succ(0) = {0}" by auto
ultimately have "a: {0} -> X" "b: {0} -> X" by auto;
then have "a = {⟨0, a`(0)⟩}" "b = {⟨0, b`(0)⟩}" using func_singleton_pair
by auto;
with A3 show "a=b" by simp;
next
let ?a = "{⟨0,x⟩}"
have "?a : {0} -> {x}" using singleton_fun by simp;
moreover from A1 A2 have "{x} ⊆ X" by simp;
ultimately have "?a : {0} -> X"
using func1_1_L1B by blast;
moreover have "{0} = succ(0)" by auto;
ultimately have "?a : succ(0) -> X" by simp;
with A1 show
"∃ a. a: succ(0) -> X ∧ a`(0) = x ∧ (∀k∈0. a`(succ(k)) = f`(a`(k)))"
using singleton_apply by auto;
qed;

text{*A lemma about restricting finite sequences needed for the proof of
the inductive step of the existence and uniqueness of finite inductive seqences.*}


lemma indseq_restrict:
assumes A1: "f: X->X" and A2: "x∈X" and A3: "n ∈ nat" and
A4: "a: succ(succ(n))-> X ∧ a`(0) = x ∧ (∀k∈succ(n). a`(succ(k)) = f`(a`(k)))"
and A5: "ar = restrict(a,succ(n))"
shows
"ar: succ(n) -> X ∧ ar`(0) = x ∧ ( ∀k∈n. ar`(succ(k)) = f`(ar`(k)) )"
proof -
from A3 have "succ(n) ⊆ succ(succ(n))" by auto;
with A4 A5 have "ar: succ(n) -> X" using restrict_type2 by auto;
moreover
from A3 have "0 ∈ succ(n)" using empty_in_every_succ by simp;
with A4 A5 have "ar`(0) = x" using restrict_if by simp;
moreover from A3 A4 A5 have "∀k∈n. ar`(succ(k)) = f`(ar`(k))"
using succ_ineq restrict_if by auto;
ultimately show ?thesis by simp;
qed;


text{*Existence and uniqueness of finite inductive sequences. The proof
is by induction and the next lemma is the inductive step.*}


lemma indseq_exun_ind:
assumes A1: "f: X->X" and A2: "x∈X" and A3: "n ∈ nat" and
A4: "∃! a. a: succ(n) -> X ∧ a`(0) = x ∧ (∀k∈n. a`(succ(k)) = f`(a`(k)))"
shows
"∃! a. a: succ(succ(n)) -> X ∧ a`(0) = x ∧
( ∀k∈succ(n). a`(succ(k)) = f`(a`(k)) )"

proof
fix a b assume
A5: "a: succ(succ(n)) -> X ∧ a`(0) = x ∧
( ∀k∈succ(n). a`(succ(k)) = f`(a`(k)) )"
and
A6: "b: succ(succ(n)) -> X ∧ b`(0) = x ∧
( ∀k∈succ(n). b`(succ(k)) = f`(b`(k)) )"

show "a = b"
proof -
let ?ar = "restrict(a,succ(n))"
let ?br = "restrict(b,succ(n))"
note A1 A2 A3 A5
moreover have "?ar = restrict(a,succ(n))" by simp;
ultimately have I:
"?ar: succ(n) -> X ∧ ?ar`(0) = x ∧ ( ∀k∈n. ?ar`(succ(k)) = f`(?ar`(k)) )"
by (rule indseq_restrict);
note A1 A2 A3 A6
moreover have "?br = restrict(b,succ(n))" by simp;
ultimately have
"?br: succ(n) -> X ∧ ?br`(0) = x ∧ ( ∀k∈n. ?br`(succ(k)) = f`(?br`(k)) )"
by (rule indseq_restrict);
with A4 I have II: "?ar = ?br" by blast;
from A3 have "succ(n) ∈ nat" by simp;
moreover from A5 A6 have
"a: succ(succ(n)) -> X" and "b: succ(succ(n)) -> X"
by auto;
moreover note II
moreover
have T: "n ∈ succ(n)" by simp;
then have "?ar`(n) = a`(n)" and "?br`(n) = b`(n)" using restrict
by auto;
with A5 A6 II T have "a`(succ(n)) = b`(succ(n))" by simp;
ultimately show "a = b" by (rule finseq_restr_eq)
qed;
next show
"∃ a. a: succ(succ(n)) -> X ∧ a`(0) = x ∧
( ∀k∈succ(n). a`(succ(k)) = f`(a`(k)) )"

proof -
from A4 obtain a where III: "a: succ(n) -> X" and IV: "a`(0) = x"
and V: "∀k∈n. a`(succ(k)) = f`(a`(k))" by auto;
let ?b = "a ∪ {⟨succ(n), f`(a`(n))⟩}"
from A1 III have
VI: "?b : succ(succ(n)) -> X" and
VII: "∀k ∈ succ(n). ?b`(k) = a`(k)" and
VIII: "?b`(succ(n)) = f`(a`(n))"
using apply_funtype finseq_extend by auto;
from A3 have "0 ∈ succ(n)" using empty_in_every_succ by simp;
with IV VII have IX: "?b`(0) = x" by auto;
{ fix k assume "k ∈ succ(n)"
then have "k∈n ∨ k = n" by auto;
moreover
{ assume A7: "k ∈ n"
with A3 VII have "?b`(succ(k)) = a`(succ(k))"
using succ_ineq by auto;
also from A7 V VII have "a`(succ(k)) = f`(?b`(k))" by simp;
finally have "?b`(succ(k)) = f`(?b`(k))" by simp }
moreover
{ assume A8: "k = n"
with VIII have "?b`(succ(k)) = f`(a`(k))" by simp
with A8 VII VIII have "?b`(succ(k)) = f`(?b`(k))" by simp }
ultimately have "?b`(succ(k)) = f`(?b`(k))" by auto;
} then have "∀k ∈ succ(n). ?b`(succ(k)) = f`(?b`(k))" by simp;
with VI IX show ?thesis by auto;
qed;
qed;

text{*The next lemma combines @{text "indseq_exun0"} and @{text "indseq_exun_ind"}
to show the existence and uniqueness of finite sequences defined by induction.*}


lemma indseq_exun:
assumes A1: "f: X->X" and A2: "x∈X" and A3: "n ∈ nat"
shows
"∃! a. a: succ(n) -> X ∧ a`(0) = x ∧ (∀k∈n. a`(succ(k)) = f`(a`(k)))"
proof -
note A3
moreover from A1 A2 have
"∃! a. a: succ(0) -> X ∧ a`(0) = x ∧ ( ∀k∈0. a`(succ(k)) = f`(a`(k)) )"
using indseq_exun0 by simp;
moreover from A1 A2 have "∀k ∈ nat.
( ∃! a. a: succ(k) -> X ∧ a`(0) = x ∧
( ∀i∈k. a`(succ(i)) = f`(a`(i)) )) -->
( ∃! a. a: succ(succ(k)) -> X ∧ a`(0) = x ∧
( ∀i∈succ(k). a`(succ(i)) = f`(a`(i)) ) )"

using indseq_exun_ind by simp;
ultimately show
"∃! a. a: succ(n) -> X ∧ a`(0) = x ∧ ( ∀k∈n. a`(succ(k)) = f`(a`(k)) )"
by (rule ind_on_nat);
qed;

text{*We are now ready to prove the main theorem about finite inductive sequences.*}

theorem fin_indseq_props:
assumes A1: "f: X->X" and A2: "x∈X" and A3: "n ∈ nat" and
A4: "a = InductiveSequenceN(x,f,n)"
shows
"a: succ(n) -> X"
"a`(0) = x"
"∀k∈n. a`(succ(k)) = f`(a`(k))"
proof -
let ?i = "THE a. a: succ(n) -> X ∧ a`(0) = x ∧
( ∀k∈n. a`(succ(k)) = f`(a`(k)) )"

from A1 A2 A3 have
"∃! a. a: succ(n) -> X ∧ a`(0) = x ∧ ( ∀k∈n. a`(succ(k)) = f`(a`(k)) )"
using indseq_exun by simp;
then have
"?i: succ(n) -> X ∧ ?i`(0) = x ∧ ( ∀k∈n. ?i`(succ(k)) = f`(?i`(k)) )"
by (rule theI);
moreover from A1 A4 have "a = ?i"
using InductiveSequenceN_def func1_1_L1 by simp;
ultimately show
"a: succ(n) -> X" "a`(0) = x" "∀k∈n. a`(succ(k)) = f`(a`(k))"
by auto;
qed;

text{*A corollary about the domain of a finite inductive sequence.*}

corollary fin_indseq_domain:
assumes A1: "f: X->X" and A2: "x∈X" and A3: "n ∈ nat"
shows "domain(InductiveSequenceN(x,f,n)) = succ(n)"
proof -
from assms have "InductiveSequenceN(x,f,n) : succ(n) -> X"
using fin_indseq_props by simp;
then show ?thesis using func1_1_L1 by simp;
qed;

text{*The collection of finite sequences defined by induction is consistent
in the sense that the restriction of the sequence defined on a larger
set to the smaller set is the same as the sequence defined on the smaller set.*}


lemma indseq_consistent: assumes A1: "f: X->X" and A2: "x∈X" and
A3: "i ∈ nat" "j ∈ nat" and A4: "i ⊆ j"
shows
"restrict(InductiveSequenceN(x,f,j),succ(i)) = InductiveSequenceN(x,f,i)"
proof -
let ?a = "InductiveSequenceN(x,f,j)"
let ?b = "restrict(InductiveSequenceN(x,f,j),succ(i))"
let ?c = "InductiveSequenceN(x,f,i)"
from A1 A2 A3 have
"?a: succ(j) -> X" "?a`(0) = x" "∀k∈j. ?a`(succ(k)) = f`(?a`(k))"
using fin_indseq_props by auto;
with A3 A4 have
"?b: succ(i) -> X ∧ ?b`(0) = x ∧ ( ∀k∈i. ?b`(succ(k)) = f`(?b`(k)))"
using succ_subset restrict_type2 empty_in_every_succ restrict succ_ineq
by auto;
moreover from A1 A2 A3 have
"?c: succ(i) -> X ∧ ?c`(0) = x ∧ ( ∀k∈i. ?c`(succ(k)) = f`(?c`(k)))"
using fin_indseq_props by simp;
moreover from A1 A2 A3 have
"∃! a. a: succ(i) -> X ∧ a`(0) = x ∧ ( ∀k∈i. a`(succ(k)) = f`(a`(k)) )"
using indseq_exun by simp;
ultimately show "?b = ?c" by blast;
qed;

text{*For any two natural numbers one of the corresponding inductive sequences
is contained in the other.*}


lemma indseq_subsets: assumes A1: "f: X->X" and A2: "x∈X" and
A3: "i ∈ nat" "j ∈ nat" and
A4: "a = InductiveSequenceN(x,f,i)" "b = InductiveSequenceN(x,f,j)"
shows "a ⊆ b ∨ b ⊆ a"
proof -
from A3 have "i⊆j ∨ j⊆i" using nat_incl_total by simp;
moreover
{ assume "i⊆j"
with A1 A2 A3 A4 have "restrict(b,succ(i)) = a"
using indseq_consistent by simp;
moreover have "restrict(b,succ(i)) ⊆ b"
using restrict_subset by simp;
ultimately have "a ⊆ b ∨ b ⊆ a" by simp; }
moreover
{ assume "j⊆i"
with A1 A2 A3 A4 have "restrict(a,succ(j)) = b"
using indseq_consistent by simp;
moreover have "restrict(a,succ(j)) ⊆ a"
using restrict_subset by simp;
ultimately have "a ⊆ b ∨ b ⊆ a" by simp }
ultimately show "a ⊆ b ∨ b ⊆ a" by auto;
qed;

text{*The first theorem about properties of infinite inductive sequences:
inductive sequence is a indeed a sequence (i.e. a function on the set of
natural numbers. *}


theorem indseq_seq: assumes A1: "f: X->X" and A2: "x∈X"
shows "InductiveSequence(x,f) : nat -> X"
proof -
let ?S = "{InductiveSequenceN(x,f,n). n ∈ nat}"
{ fix a assume "a∈?S"
then obtain n where "n ∈ nat" and "a = InductiveSequenceN(x,f,n)"
by auto;
with A1 A2 have "a : succ(n)->X" using fin_indseq_props
by simp;
then have "∃A B. a:A->B" by auto
} then have "∀a ∈ ?S. ∃A B. a:A->B" by auto;
moreover
{ fix a b assume "a∈?S" "b∈?S"
then obtain i j where "i∈nat" "j∈nat" and
"a = InductiveSequenceN(x,f,i)" "b = InductiveSequenceN(x,f,j)"
by auto;
with A1 A2 have "a⊆b ∨ b⊆a" using indseq_subsets by simp;
} then have "∀a∈?S. ∀b∈?S. a⊆b ∨ b⊆a" by auto;
ultimately have "\<Union>?S : domain(\<Union>?S) -> range(\<Union>?S)"
using fun_Union by simp;
with A1 A2 have I: "\<Union>?S : nat -> range(\<Union>?S)"
using domain_UN fin_indseq_domain nat_union_succ by simp
moreover
{ fix k assume A3: "k ∈ nat"
let ?y = "(\<Union>?S)`(k)"
note I A3
moreover have "?y = (\<Union>?S)`(k)" by simp;
ultimately have "⟨k,?y⟩ ∈ (\<Union>?S)" by (rule func1_1_L5A);
then obtain n where "n ∈ nat" and II: "⟨k,?y⟩ ∈ InductiveSequenceN(x,f,n)"
by auto;
with A1 A2 have "InductiveSequenceN(x,f,n): succ(n) -> X"
using fin_indseq_props by simp;
with II have "?y ∈ X" using func1_1_L5 by blast;
} then have "∀k ∈ nat. (\<Union>?S)`(k) ∈ X" by simp;
ultimately have "\<Union>?S : nat -> X" using func1_1_L1A
by blast;
then show "InductiveSequence(x,f) : nat -> X"
using InductiveSequence_def by simp;
qed;

text{*Restriction of an inductive sequence to a finite domain
is the corresponding finite inductive sequence.*}


lemma indseq_restr_eq:
assumes A1: "f: X->X" and A2: "x∈X" and A3: "n ∈ nat"
shows
"restrict(InductiveSequence(x,f),succ(n)) = InductiveSequenceN(x,f,n)"
proof -
let ?a = "InductiveSequence(x,f)"
let ?b = "InductiveSequenceN(x,f,n)"
let ?S = "{InductiveSequenceN(x,f,n). n ∈ nat}"
from A1 A2 A3 have
I: "?a : nat -> X" and "succ(n) ⊆ nat"
using indseq_seq succnat_subset_nat by auto;
then have "restrict(?a,succ(n)) : succ(n) -> X"
using restrict_type2 by simp;
moreover from A1 A2 A3 have "?b : succ(n) -> X"
using fin_indseq_props by simp;
moreover
{ fix k assume A4: "k ∈ succ(n)"
from A1 A2 A3 I have
"\<Union>?S : nat -> X" "?b ∈ ?S" "?b : succ(n) -> X"
using InductiveSequence_def fin_indseq_props by auto;
with A4 have "restrict(?a,succ(n))`(k) = ?b`(k)"
using fun_Union_apply InductiveSequence_def restrict_if
by simp;
} then have "∀k ∈ succ(n). restrict(?a,succ(n))`(k) = ?b`(k)"
by simp;
ultimately show ?thesis by (rule func_eq);
qed;

text{*The first element of the inductive sequence starting at $x$ and generated by $f$
is indeed $x$. *}


theorem indseq_valat0: assumes A1: "f: X->X" and A2: "x∈X"
shows "InductiveSequence(x,f)`(0) = x"
proof -
let ?a = "InductiveSequence(x,f)"
let ?b = "InductiveSequenceN(x,f,0)"
have T: "0∈nat" "0 ∈ succ(0)" by auto;
with A1 A2 have "?b`(0) = x"
using fin_indseq_props by simp;
moreover from T have "restrict(?a,succ(0))`(0) = ?a`(0)"
using restrict_if by simp;
moreover from A1 A2 T have
"restrict(?a,succ(0)) = ?b"
using indseq_restr_eq by simp;
ultimately show "?a`(0) = x" by simp;
qed;

text{*An infinite inductive sequence satisfies the
inductive relation that defines it.*}


theorem indseq_vals:
assumes A1: "f: X->X" and A2: "x∈X" and A3: "n ∈ nat"
shows
"InductiveSequence(x,f)`(succ(n)) = f`(InductiveSequence(x,f)`(n))"
proof -
let ?a = "InductiveSequence(x,f)"
let ?b = "InductiveSequenceN(x,f,succ(n))"
from A3 have T:
"succ(n) ∈ succ(succ(n))"
"succ(succ(n)) ∈ nat"
"n ∈ succ(succ(n))"
by auto;
then have "?a`(succ(n)) = restrict(?a,succ(succ(n)))`(succ(n))"
using restrict_if by simp;
also from A1 A2 T have "… = f`(restrict(?a,succ(succ(n)))`(n))"
using indseq_restr_eq fin_indseq_props by simp;
also from T have "… = f`(?a`(n))" using restrict_if by simp;
finally show "?a`(succ(n)) = f`(?a`(n))" by simp;
qed;

section{*Images of inductive sequences*}

text{*In this section we consider the properties of sets that are
images of inductive sequences, that is are of the form
$\{f^{(n)} (x) : n \in N\}$ for some $x$ in the domain of $f$,
where $f^{(n)}$ denotes the $n$'th
iteration of the function $f$. For a function $f:X\rightarrow X$ and
a point $x\in X$ such set is set is sometimes called
the orbit of $x$ generated by $f$.*}


text{*The basic properties of orbits.*}

theorem ind_seq_image: assumes A1: "f: X->X" and A2: "x∈X" and
A3: "A = InductiveSequence(x,f)``(nat)"
shows "x∈A" and "∀y∈A. f`(y) ∈ A"
proof -
let ?a = "InductiveSequence(x,f)"
from A1 A2 have "?a : nat -> X" using indseq_seq
by simp;
with A3 have I: "A = {?a`(n). n ∈ nat}" using func_imagedef
by auto hence "?a`(0) ∈ A" by auto
with A1 A2 show "x∈A" using indseq_valat0 by simp;
{ fix y assume "y∈A"
with I obtain n where II: "n ∈ nat" and III: "y = ?a`(n)"
by auto;
with A1 A2 have "?a`(succ(n)) = f`(y)"
using indseq_vals by simp;
moreover from I II have "?a`(succ(n)) ∈ A" by auto;
ultimately have "f`(y) ∈ A" by simp;
} then show "∀y∈A. f`(y) ∈ A" by simp;
qed;

section{*Subsets generated by a binary operation*}

text{* In algebra we often talk about sets "generated" by an element,
that is sets of the form (in multiplicative notation) $\{a^n | n\in Z\}$.
This is a related to a general notion of "power"
(as in $a^n = a\cdot a \cdot .. \cdot a$ ) or multiplicity $n\cdot a = a+a+..+a$.
The intuitive meaning of such notions is obvious, but we need to do some
work to be able to use it in the formalized setting. This sections is devoted
to sequences that are created by repeatedly applying a binary operation with the
second argument fixed to some constant.*}


text{*Basic propertes of sets generated by binary operations.*}

theorem binop_gen_set:
assumes A1: "f: X×Y -> X" and A2: "x∈X" "y∈Y" and
A3: "a = InductiveSequence(x,Fix2ndVar(f,y))"
shows
"a : nat -> X"
"a``(nat) ∈ Pow(X)"
"x ∈ a``(nat)"
"∀z ∈ a``(nat). Fix2ndVar(f,y)`(z) ∈ a``(nat)"
proof -
let ?g = "Fix2ndVar(f,y)"
from A1 A2 have I: "?g : X->X"
using fix_2nd_var_fun by simp;
with A2 A3 show "a : nat -> X"
using indseq_seq by simp;
then show "a``(nat) ∈ Pow(X)" using func1_1_L6 by simp;
from A2 A3 I show "x ∈ a``(nat)" using ind_seq_image by blast;
from A2 A3 I have
"?g : X->X" "x∈X" "a``(nat) = InductiveSequence(x,?g)``(nat)"
by auto;
then show "∀z ∈ a``(nat). Fix2ndVar(f,y)`(z) ∈ a``(nat)"
by (rule ind_seq_image);
qed;

text{*A simple corollary to the theorem @{text "binop_gen_set"}: a set
that contains all iterations of the application of a binary operation
exists.*}


lemma binop_gen_set_ex: assumes A1: "f: X×Y -> X" and A2: "x∈X" "y∈Y"
shows "{A ∈ Pow(X). x∈A ∧ (∀z ∈ A. f`⟨z,y⟩ ∈ A) } ≠ 0"
proof -
let ?a = "InductiveSequence(x,Fix2ndVar(f,y))"
let ?A = "?a``(nat)"
from A1 A2 have I: "?A ∈ Pow(X)" and "x ∈ ?A" using binop_gen_set
by auto;
moreover
{ fix z assume T: "z∈?A"
with A1 A2 have "Fix2ndVar(f,y)`(z) ∈ ?A"
using binop_gen_set by simp;
moreover
from I T have "z ∈ X" by auto;
with A1 A2 have "Fix2ndVar(f,y)`(z) = f`⟨z,y⟩"
using fix_var_val by simp;
ultimately have "f`⟨z,y⟩ ∈ ?A" by simp;
} then have "∀z ∈ ?A. f`⟨z,y⟩ ∈ ?A" by simp;
ultimately show ?thesis by auto;
qed;

text{*A more general version of @{text " binop_gen_set"} where the generating
binary operation acts on a larger set.*}


theorem binop_gen_set1: assumes A1: "f: X×Y -> X" and
A2: "X1 ⊆ X" and A3: "x∈X1" "y∈Y" and
A4: "∀t∈X1. f`⟨t,y⟩ ∈ X1" and
A5: "a = InductiveSequence(x,Fix2ndVar(restrict(f,X1×Y),y))"
shows
"a : nat -> X1"
"a``(nat) ∈ Pow(X1)"
"x ∈ a``(nat)"
"∀z ∈ a``(nat). Fix2ndVar(f,y)`(z) ∈ a``(nat)"
"∀z ∈ a``(nat). f`⟨z,y⟩ ∈ a``(nat)"
proof -
let ?h = "restrict(f,X1×Y)"
let ?g = "Fix2ndVar(?h,y)"
from A2 have "X1×Y ⊆ X×Y" by auto;
with A1 have I: "?h : X1×Y -> X"
using restrict_type2 by simp;
with A3 have II: "?g: X1 -> X" using fix_2nd_var_fun by simp;
from A3 A4 I have "∀t∈X1. ?g`(t) ∈ X1"
using restrict fix_var_val by simp;
with II have III: "?g : X1 -> X1" using func1_1_L1A by blast;
with A3 A5 show "a : nat -> X1" using indseq_seq by simp;
then show IV: "a``(nat) ∈ Pow(X1)" using func1_1_L6 by simp;
from A3 A5 III show "x ∈ a``(nat)" using ind_seq_image by blast;
from A3 A5 III have
"?g : X1 -> X1" "x∈X1" "a``(nat) = InductiveSequence(x,?g)``(nat)"
by auto;
then have "∀z ∈ a``(nat). Fix2ndVar(?h,y)`(z) ∈ a``(nat)"
by (rule ind_seq_image);
moreover
{ fix z assume "z ∈ a``(nat)"
with IV have "z ∈ X1" by auto;
with A1 A2 A3 have "?g`(z) = Fix2ndVar(f,y)`(z)"
using fix_2nd_var_restr_comm restrict by simp;
} then have "∀z ∈ a``(nat). ?g`(z) = Fix2ndVar(f,y)`(z)" by simp;
ultimately show "∀z ∈ a``(nat). Fix2ndVar(f,y)`(z) ∈ a``(nat)" by simp;
moreover
{ fix z assume "z ∈ a``(nat)"
with A2 IV have "z∈X" by auto;
with A1 A3 have "Fix2ndVar(f,y)`(z) = f`⟨z,y⟩"
using fix_var_val by simp;
} then have "∀z ∈ a``(nat). Fix2ndVar(f,y)`(z) = f`⟨z,y⟩"
by simp;
ultimately show "∀z ∈ a``(nat). f`⟨z,y⟩ ∈ a``(nat)"
by simp;
qed;

text{*A generalization of @{text " binop_gen_set_ex"} that applies when the binary
operation acts on a larger set. This is used in our Metamath translation
to prove the existence of the set of real natural numbers.
Metamath defines the real natural numbers as the smallest set that cantains
$1$ and is closed with respect to operation of adding $1$.*}


lemma binop_gen_set_ex1: assumes A1: "f: X×Y -> X" and
A2: "X1 ⊆ X" and A3: "x∈X1" "y∈Y" and
A4: "∀t∈X1. f`⟨t,y⟩ ∈ X1"
shows "{A ∈ Pow(X1). x∈A ∧ (∀z ∈ A. f`⟨z,y⟩ ∈ A) } ≠ 0"
proof -
let ?a = "InductiveSequence(x,Fix2ndVar(restrict(f,X1×Y),y))"
let ?A = "?a``(nat)"
from A1 A2 A3 A4 have
"?A ∈ Pow(X1)" "x ∈ ?A" "∀z ∈ ?A. f`⟨z,y⟩ ∈ ?A"
using binop_gen_set1 by auto;
thus ?thesis by auto;
qed;

section{*Inductive sequences with changing generating function*}

text{*A seemingly more general form of a sequence defined by induction
is a sequence generated by the difference equation $x_{n+1} = f_{n} (x_n)$
where $n\mapsto f_n$ is a given sequence of functions such
that each maps $X$ into inself.
For example when
$f_n (x) := x + x_n$ then the equation $S_{n+1} = f_{n} (S_n)$ describes
the sequence $n \mapsto S_n = s_0 +\sum_{i=0}^n x_n$, i.e. the sequence of
partial sums of the sequence $\{s_0, x_0, x_1, x_3,..\}$.
*}


text{*The situation where the function that we iterate changes with $n$ can be
derived from the simpler case if we define the generating function appropriately.
Namely, we replace the generating function in the definitions of
@{text "InductiveSequenceN"} by the function $f: X\times n \rightarrow X\times n$,
$f\langle x,k\rangle = \langle f_k(x), k+1 \rangle$ if $k < n$,
$\langle f_k(x), k \rangle$ otherwise. The first notion defines the expression
we will use to define the generating function.
To understand the notation recall that in standard Isabelle/ZF
for a pair $s=\langle x,n \rangle$ we have @{text "fst"}$(s)=x$ and
@{text "snd"}$(s)=n$. *}


definition
"StateTransfFunNMeta(F,n,s) ≡
if (snd(s) ∈ n) then ⟨F`(snd(s))`(fst(s)), succ(snd(s))⟩ else s"


text{* Then we define the actual generating function on sets of pairs
from $X\times \{0,1, .. ,n\}$. *}


definition
"StateTransfFunN(X,F,n) ≡ {⟨s, StateTransfFunNMeta(F,n,s)⟩. s ∈ X×succ(n)}"

text{*Having the generating function we can define the expression
that we cen use to define the inductive sequence generates.*}


definition
"StatesSeq(x,X,F,n) ≡
InductiveSequenceN(⟨x,0⟩, StateTransfFunN(X,F,n),n)"


text{*Finally we can define the sequence given by a initial point $x$,
and a sequence $F$ of $n$ functions.*}


definition
"InductiveSeqVarFN(x,X,F,n) ≡ {⟨k,fst(StatesSeq(x,X,F,n)`(k))⟩. k ∈ succ(n)}"

text{*The state transformation function (@{text "StateTransfFunN"} is
a function that transforms $X\times n$ into itself.*}


lemma state_trans_fun: assumes A1: "n ∈ nat" and A2: "F: n -> (X->X)"
shows "StateTransfFunN(X,F,n): X×succ(n) -> X×succ(n)"
proof -
{ fix s assume A3: "s ∈ X×succ(n)"
let ?x = "fst(s)"
let ?k = "snd(s)"
let ?S = "StateTransfFunNMeta(F,n,s)"
from A3 have T: "?x ∈ X" "?k ∈ succ(n)" and "⟨?x,?k⟩ = s" by auto;
{ assume A4: "?k ∈ n"
with A1 have "succ(?k) ∈ succ(n)" using succ_ineq by simp;
with A2 T A4 have "?S ∈ X×succ(n)"
using apply_funtype StateTransfFunNMeta_def by simp }
with A2 A3 T have "?S ∈ X×succ(n)"
using apply_funtype StateTransfFunNMeta_def by auto;
} then have "∀s ∈ X×succ(n). StateTransfFunNMeta(F,n,s) ∈ X×succ(n)"
by simp;
then have
"{⟨s, StateTransfFunNMeta(F,n,s)⟩. s ∈ X×succ(n)} : X×succ(n) -> X×succ(n)"
by (rule ZF_fun_from_total);
then show "StateTransfFunN(X,F,n): X×succ(n) -> X×succ(n)"
using StateTransfFunN_def by simp;
qed;

text{*We can apply @{text "fin_indseq_props"} to the sequence used in the
definition of @{text "InductiveSeqVarFN"} to get the properties of the sequence
of states generated by the @{text "StateTransfFunN"}.*}


lemma states_seq_props:
assumes A1: "n ∈ nat" and A2: "F: n -> (X->X)" and A3: "x∈X" and
A4: "b = StatesSeq(x,X,F,n)"
shows
"b : succ(n) -> X×succ(n)"
"b`(0) = ⟨x,0⟩"
"∀k ∈ succ(n). snd(b`(k)) = k"
"∀k∈n. b`(succ(k)) = ⟨F`(k)`(fst(b`(k))), succ(k)⟩"
proof -
let ?f = "StateTransfFunN(X,F,n)"
from A1 A2 have I: "?f : X×succ(n) -> X×succ(n)"
using state_trans_fun by simp
moreover from A1 A3 have II: "⟨x,0⟩ ∈ X×succ(n)"
using empty_in_every_succ by simp;
moreover note A1
moreover from A4 have III: "b = InductiveSequenceN(⟨x,0⟩,?f,n)"
using StatesSeq_def by simp;
ultimately show IV: "b : succ(n) -> X×succ(n)"
by (rule fin_indseq_props);
from I II A1 III show V: "b`(0) = ⟨x,0⟩"
by (rule fin_indseq_props);
from I II A1 III have VI: "∀k∈n. b`(succ(k)) = ?f`(b`(k))"
by (rule fin_indseq_props)
{ fix k
note I
moreover
assume A5: "k ∈ n" hence "k ∈ succ(n)" by auto;
with IV have "b`(k) ∈ X×succ(n)" using apply_funtype by simp;
moreover have "?f = {⟨s, StateTransfFunNMeta(F,n,s)⟩. s ∈ X×succ(n)}"
using StateTransfFunN_def by simp;
ultimately have "?f`(b`(k)) = StateTransfFunNMeta(F,n,b`(k))"
by (rule ZF_fun_from_tot_val);
} then have VII: "∀k ∈ n. ?f`(b`(k)) = StateTransfFunNMeta(F,n,b`(k))"
by simp;
{ fix k assume A5: "k ∈ succ(n)"
note A1 A5
moreover from V have " snd(b`(0)) = 0" by simp;
moreover from VI VII have
"∀j∈n. snd(b`(j)) = j --> snd(b`(succ(j))) = succ(j)"
using StateTransfFunNMeta_def by auto;
ultimately have "snd(b`(k)) = k" by (rule fin_nat_ind);
} then show "∀k ∈ succ(n). snd(b`(k)) = k" by simp;
with VI VII show "∀k∈n. b`(succ(k)) = ⟨F`(k)`(fst(b`(k))), succ(k)⟩"
using StateTransfFunNMeta_def by auto;
qed;

text{*Basic properties of sequences defined by equation $x_{n+1}=f_n (x_n)$.*}

theorem fin_indseq_var_f_props:
assumes A1: "n ∈ nat" and A2: "x∈X" and A3: "F: n -> (X->X)" and
A4: "a = InductiveSeqVarFN(x,X,F,n)"
shows
"a: succ(n) -> X"
"a`(0) = x"
"∀k∈n. a`(succ(k)) = F`(k)`(a`(k))"
proof -
let ?f = "StateTransfFunN(X,F,n)"
let ?b = "StatesSeq(x,X,F,n)"
from A1 A2 A3 have "?b : succ(n) -> X×succ(n)"
using states_seq_props by simp;
then have "∀k ∈ succ(n). ?b`(k) ∈ X×succ(n)"
using apply_funtype by simp;
hence "∀k ∈ succ(n). fst(?b`(k)) ∈ X" by auto;
then have I: "{⟨k,fst(?b`(k))⟩. k ∈ succ(n)} : succ(n) -> X"
by (rule ZF_fun_from_total);
with A4 show II: "a: succ(n) -> X" using InductiveSeqVarFN_def
by simp;
moreover from A1 have "0 ∈ succ(n)" using empty_in_every_succ
by simp;
moreover from A4 have III:
"a = {⟨k,fst(StatesSeq(x,X,F,n)`(k))⟩. k ∈ succ(n)}"
using InductiveSeqVarFN_def by simp;
ultimately have "a`(0) = fst(?b`(0))"
by (rule ZF_fun_from_tot_val);
with A1 A2 A3 show "a`(0) = x" using states_seq_props by auto;
{ fix k
assume A5: "k ∈ n"
with A1 have T1: "succ(k) ∈ succ(n)" and T2: "k ∈ succ(n)"
using succ_ineq by auto;
from II T1 III have "a`(succ(k)) = fst(?b`(succ(k)))"
by (rule ZF_fun_from_tot_val);
with A1 A2 A3 A5 have "a`(succ(k)) = F`(k)`(fst(?b`(k)))"
using states_seq_props by simp;
moreover from II T2 III have "a`(k) = fst(?b`(k))"
by (rule ZF_fun_from_tot_val);
ultimately have "a`(succ(k)) = F`(k)`(a`(k))"
by simp;
} then show "∀k∈n. a`(succ(k)) = F`(k)`(a`(k))"
by simp;
qed;

text{*A consistency condition: if we make the sequence of
generating functions shorter, then we get a shorter inductive
sequence with the same values as in the original sequence. *}


lemma fin_indseq_var_f_restrict: assumes
A1: "n ∈ nat" "i ∈ nat" "x∈X" "F: n -> (X->X)" "G: i -> (X->X)"
and A2: "i ⊆ n" and A3: "∀j∈i. G`(j) = F`(j)" and A4: "k ∈ succ(i)"
shows "InductiveSeqVarFN(x,X,G,i)`(k) = InductiveSeqVarFN(x,X,F,n)`(k)"
proof -
let ?a = "InductiveSeqVarFN(x,X,F,n)"
let ?b = "InductiveSeqVarFN(x,X,G,i)"
from A1 A4 have "i ∈ nat" "k ∈ succ(i)" by auto;
moreover from A1 have "?b`(0) = ?a`(0)"
using fin_indseq_var_f_props by simp;
moreover from A1 A2 A3 have
"∀j∈i. ?b`(j) = ?a`(j) --> ?b`(succ(j)) = ?a`(succ(j))"
using fin_indseq_var_f_props by auto;
ultimately show "?b`(k) = ?a`(k)"
by (rule fin_nat_ind);
qed;


(*text{*If we make the sequence of generating funtions shorter, the resulting
sequence will be shorter.*}

lemma fin_indseq_var_f_restrict:
assumes A1: "n ∈ nat" and A2: "x∈X" and
A3: "F: n -> (X->X)" and A4: "k ≤ n" and
A5: "a = restrict(InductiveSeqVarFN(x,X,F,n),k)" and
A6: "b = InductiveSeqVarFN(x,X,restrict(F,k),k)"
shows "a = b"
proof -*)


end