IsarMathLib

Proofs by humans, for humans, formally verified by Isabelle/ZF proof assistant

theory Nat_ZF_IML imports ZF.Arith

begin

The ZF set theory constructs natural numbers from the empty set and the notion of a one-element set. Namely, zero of natural numbers is defined as the empty set. For each natural number \(n\) the next natural number is defined as \(n\cup \{n\}\). With this definition for every non-zero natural number we get the identity \(n = \{0,1,2,..,n-1\}\). It is good to remember that when we see an expression like \(f: n \rightarrow X\). Also, with this definition the relation "less or equal than" becomes "\(\subseteq\)" and the relation "less than" becomes "\(\in\)".

Induction

The induction lemmas in the standard Isabelle's Nat.thy file like for example nat_induct require the induction step to be a higher order statement (the one that uses the \(\Longrightarrow\) sign). I found it difficult to apply from Isar, which is perhaps more of an indication of my Isar skills than anything else. Anyway, here we provide a first order version that is easier to reference in Isar declarative style proofs.

The next theorem is a version of induction on natural numbers that I was thought in school.

theorem ind_on_nat:

assumes A1: \( n\in nat \) and A2: \( P(0) \) and A3: \( \forall k\in nat.\ P(k)\longrightarrow P(succ(k)) \)

shows \( P(n) \)proof

note A1, A2

moreover {

fix \( x \)

assume \( x\in nat \), \( P(x) \)

with A3 have \( P(succ(x)) \)

} ultimately show \( P(n) \) by (rule nat_induct )

qed

A nonzero natural number has a predecessor.

lemma Nat_ZF_1_L3:

assumes A1: \( n \in nat \) and A2: \( n\neq 0 \)

shows \( \exists k\in nat.\ n = succ(k) \)proof

from A1 have \( n \in \{0\} \cup \{succ(k).\ k\in nat\} \) using nat_unfold

with A2 show \( thesis \)

qed

What is succ, anyway? It's a union with the singleton of the set.

lemma succ_explained:

shows \( succ(n) = n \cup \{n\} \) using succ_iff

Empty set is an element of every natural number which is not zero.

lemma empty_in_every_succ:

assumes A1: \( n \in nat \)

shows \( 0 \in succ(n) \)proof

note A1

moreover

have \( 0 \in succ(0) \)

moreover {

fix \( k \)

assume \( k \in nat \) and A2: \( 0 \in succ(k) \)

then have \( succ(k) \subseteq succ(succ(k)) \)

with A2 have \( 0 \in succ(succ(k)) \)

}

then have \( \forall k \in nat.\ 0 \in succ(k) \longrightarrow 0 \in succ(succ(k)) \)

ultimately show \( 0 \in succ(n) \) by (rule ind_on_nat )

qed

A more direct way of stating that empty set is an element of every non-zero natural number:

lemma empty_in_non_empty:

assumes \( n\in nat \), \( n\neq 0 \)

shows \( 0\in n \) using assms, Nat_ZF_1_L3, empty_in_every_succ

If one natural number is less than another then their successors are in the same relation.

lemma succ_ineq:

assumes A1: \( n \in nat \)

shows \( \forall i \in n.\ succ(i) \in succ(n) \)proof

note A1

moreover

have \( \forall k \in 0.\ succ(k) \in succ(0) \)

moreover {

fix \( k \)

assume A2: \( \forall i\in k.\ succ(i) \in succ(k) \)

{

fix \( i \)

assume \( i \in succ(k) \)

then have \( i \in k \vee i = k \)

moreover {

assume \( i\in k \)

with A2 have \( succ(i) \in succ(k) \)

hence \( succ(i) \in succ(succ(k)) \)

} moreover {

assume \( i = k \)

then have \( succ(i) \in succ(succ(k)) \)

} ultimately have \( succ(i) \in succ(succ(k)) \)

}

then have \( \forall i \in succ(k).\ succ(i) \in succ(succ(k)) \)

}

then have \( \forall k \in nat.\ \) \( ( (\forall i\in k.\ succ(i) \in succ(k)) \longrightarrow (\forall i \in succ(k).\ succ(i) \in succ(succ(k))) ) \)

ultimately show \( \forall i \in n.\ succ(i) \in succ(n) \) by (rule ind_on_nat )

qed

For natural numbers if \(k\subseteq n\) the similar holds for their successors.

lemma succ_subset:

assumes A1: \( k \in nat \), \( n \in nat \) and A2: \( k\subseteq n \)

shows \( succ(k) \subseteq succ(n) \)proof

from A1 have T: \( Ord(k) \) and \( Ord(n) \) using nat_into_Ord

with A2 have \( succ(k) \leq succ(n) \) using subset_imp_le

then show \( succ(k) \subseteq succ(n) \) using le_imp_subset

qed

For any two natural numbers one of them is contained in the other.

lemma nat_incl_total:

assumes A1: \( i \in nat \), \( j \in nat \)

shows \( i \subseteq j \vee j \subseteq i \)proof

from A1 have T: \( Ord(i) \), \( Ord(j) \) using nat_into_Ord

then have \( i\in j \vee i=j \vee j\in i \) using Ord_linear

moreover {

assume \( i\in j \)

with T have \( i\subseteq j \vee j\subseteq i \) using lt_def, leI, le_imp_subset

} moreover {

assume \( i=j \)

then have \( i\subseteq j \vee j\subseteq i \)

} moreover {

assume \( j\in i \)

with T have \( i\subseteq j \vee j\subseteq i \) using lt_def, leI, le_imp_subset

} ultimately show \( i \subseteq j \vee j \subseteq i \)

qed

The set of natural numbers is the union of all successors of natural numbers.

lemma nat_union_succ:

shows \( nat = (\bigcup n \in nat.\ succ(n)) \)proof

show \( nat \subseteq (\bigcup n \in nat.\ succ(n)) \)

next

{

fix \( k \)

assume A2: \( k \in (\bigcup n \in nat.\ succ(n)) \)

then obtain \( n \) where T: \( n \in nat \) and I: \( k \in succ(n) \)

then have \( k \leq n \) using nat_into_Ord, lt_def

with T have \( k \in nat \) using le_in_nat

}

then show \( (\bigcup n \in nat.\ succ(n)) \subseteq nat \)

qed

Successors of natural numbers are subsets of the set of natural numbers.

lemma succnat_subset_nat:

assumes A1: \( n \in nat \)

shows \( succ(n) \subseteq nat \)proof

from A1 have \( succ(n) \subseteq (\bigcup n \in nat.\ succ(n)) \)

then show \( succ(n) \subseteq nat \) using nat_union_succ

qed

Element of a natural number is a natural number.

lemma elem_nat_is_nat:

assumes A1: \( n \in nat \) and A2: \( k\in n \)

shows \( k \lt n \), \( k \in nat \), \( k \leq n \), \( \langle k,n\rangle \in Le \)proof

from A1, A2 show \( k \lt n \) using nat_into_Ord, lt_def

with A1 show \( k \in nat \) using lt_nat_in_nat

from \( k \lt n \) show \( k \leq n \) using leI

with A1, \( k \in nat \) show \( \langle k,n\rangle \in Le \) using Le_def

qed

For natural numbers membership and inequality are the same and \(k \leq n\) is the same as \(k \in \textrm{succ}(n)\). The proof relies on lemmas in the standard Isabelle's Nat and Ordinal theories.

lemma nat_mem_lt:

assumes \( n\in nat \)

shows \( k \lt n \longleftrightarrow k\in n \) and \( k\leq n \longleftrightarrow k \in succ(n) \) using assms, nat_into_Ord, Ord_mem_iff_lt

The term \(k \leq j\) the same as \(k < \textrm{succ}(n)\).

lemma leq_mem_succ:

shows \( k\leq n \longleftrightarrow k \lt succ(n) \)

If the successor of a natural number \(k\) is an element of the successor of \(n\) then a similar relations holds for the numbers themselves.

lemma succ_mem:

assumes \( n \in nat \), \( succ(k) \in succ(n) \)

shows \( k\in n \) using assms, elem_nat_is_nat(1), succ_leE, nat_into_Ord unfolding lt_def

The set of natural numbers is the union of its elements.

lemma nat_union_nat:

shows \( nat = \bigcup nat \) using elem_nat_is_nat

A natural number is a subset of the set of natural numbers.

lemma nat_subset_nat:

assumes A1: \( n \in nat \)

shows \( n \subseteq nat \)proof

from A1 have \( n \subseteq \bigcup nat \)

then show \( n \subseteq nat \) using nat_union_nat

qed

Adding natural numbers does not decrease what we add to.

lemma add_nat_le:

assumes A1: \( n \in nat \) and A2: \( k \in nat \)

shows \( n \leq n +_\mathbb{N} k \), \( n \subseteq n +_\mathbb{N} k \), \( n \subseteq k +_\mathbb{N} n \)proof

from A1, A2 have \( n \leq n \), \( 0 \leq k \), \( n \in nat \), \( k \in nat \) using nat_le_refl, nat_0_le

then have \( n +_\mathbb{N} 0 \leq n +_\mathbb{N} k \) by (rule add_le_mono )

with A1 show \( n \leq n +_\mathbb{N} k \) using add_0_right

then show \( n \subseteq n +_\mathbb{N} k \) using le_imp_subset

then show \( n \subseteq k +_\mathbb{N} n \) using add_commute

qed

Result of adding an element of \(k\) is smaller than of adding \(k\).

lemma add_lt_mono:

assumes \( k \in nat \) and \( j\in k \)

shows \( (n +_\mathbb{N} j) \lt (n +_\mathbb{N} k) \), \( (n +_\mathbb{N} j) \in (n +_\mathbb{N} k) \)proof

from assms have \( j \lt k \) using elem_nat_is_nat

moreover

note \( k \in nat \)

ultimately show \( (n +_\mathbb{N} j) \lt (n +_\mathbb{N} k) \), \( (n +_\mathbb{N} j) \in (n +_\mathbb{N} k) \) using add_lt_mono2, ltD

qed

A technical lemma about a decomposition of a sum of two natural numbers: if a number \(i\) is from \(m + n\) then it is either from \(m\) or can be written as a sum of \(m\) and a number from \(n\). The proof by induction w.r.t. to \(m\) seems to be a bit heavy-handed, but I could not figure out how to do this directly from results from standard Isabelle/ZF.

lemma nat_sum_decomp:

assumes A1: \( n \in nat \) and A2: \( m \in nat \)

shows \( \forall i \in m +_\mathbb{N} n.\ i \in m \vee (\exists j \in n.\ i = m +_\mathbb{N} j) \)proof

note A1

moreover

from A2 have \( \forall i \in m +_\mathbb{N} 0.\ i \in m \vee (\exists j \in 0.\ i = m +_\mathbb{N} j) \) using add_0_right

moreover

have \( \forall k\in nat.\ \) \( (\forall i \in m +_\mathbb{N} k.\ i \in m \vee (\exists j \in k.\ i = m +_\mathbb{N} j)) \longrightarrow \) \( (\forall i \in m +_\mathbb{N} succ(k).\ i \in m \vee (\exists j \in succ(k).\ i = m +_\mathbb{N} j)) \)proof

{

fix \( k \)

assume A3: \( k \in nat \)

{

assume A4: \( \forall i \in m +_\mathbb{N} k.\ i \in m \vee (\exists j \in k.\ i = m +_\mathbb{N} j) \)

{

fix \( i \)

assume \( i \in m +_\mathbb{N} succ(k) \)

then have \( i \in m +_\mathbb{N} k \vee i = m +_\mathbb{N} k \) using add_succ_right

moreover

from A4, A3 have \( i \in m +_\mathbb{N} k \longrightarrow i \in m \vee (\exists j \in succ(k).\ i = m +_\mathbb{N} j) \)

ultimately have \( i \in m \vee (\exists j \in succ(k).\ i = m +_\mathbb{N} j) \)

}

then have \( \forall i \in m +_\mathbb{N} succ(k).\ i \in m \vee (\exists j \in succ(k).\ i = m +_\mathbb{N} j) \)

}

then have \( (\forall i \in m +_\mathbb{N} k.\ i \in m \vee (\exists j \in k.\ i = m +_\mathbb{N} j)) \longrightarrow \) \( (\forall i \in m +_\mathbb{N} succ(k).\ i \in m \vee (\exists j \in succ(k).\ i = m +_\mathbb{N} j)) \)

}

then show \( thesis \)

qed

ultimately show \( \forall i \in m +_\mathbb{N} n.\ i \in m \vee (\exists j \in n.\ i = m +_\mathbb{N} j) \) by (rule ind_on_nat )

qed

A variant of induction useful for finite sequences.

lemma fin_nat_ind:

assumes A1: \( n \in nat \) and A2: \( k \in succ(n) \) and A3: \( P(0) \) and A4: \( \forall j\in n.\ P(j) \longrightarrow P(succ(j)) \)

shows \( P(k) \)proof

from A2 have \( k \in n \vee k=n \)

with A1 have \( k \in nat \) using elem_nat_is_nat

moreover

from A3 have \( 0 \in succ(n) \longrightarrow P(0) \)

moreover

from A1, A4 have \( \forall k \in nat.\ (k \in succ(n) \longrightarrow P(k)) \longrightarrow (succ(k) \in succ(n) \longrightarrow P(succ(k))) \) using nat_into_Ord, Ord_succ_mem_iff

ultimately have \( k \in succ(n) \longrightarrow P(k) \) by (rule ind_on_nat )

with A2 show \( P(k) \)

qed

Some properties of positive natural numbers.

lemma succ_plus:

assumes \( n \in nat \), \( k \in nat \)

shows \( succ(n +_\mathbb{N} j) \in nat \), \( succ(n) +_\mathbb{N} succ(j) = succ(succ(n +_\mathbb{N} j)) \) using assms

If \(k\) is in the successor of \(n\), then the predecessor of \(k\) is in \(n\).

lemma pred_succ_mem:

assumes \( n\in nat \), \( n\neq 0 \), \( k\in succ(n) \)

shows \( pred(k)\in n \)proof

from assms(1,3) have \( k\in nat \) using succnat_subset_nat

{

assume \( k\neq 0 \)

with \( k\in nat \) obtain \( j \) where \( j\in nat \) and \( k=succ(j) \) using Nat_ZF_1_L3

with assms(1,3) have \( pred(k)\in n \) using succ_mem, pred_succ_eq

}

moreover {

assume \( k=0 \)

with assms(1,2) have \( pred(k)\in n \) using pred_0, empty_in_non_empty

} ultimately show \( thesis \)

qed

For non-zero natural numbers \(\textrm{pred}(n) = n-1\).

lemma pred_minus_one:

assumes \( n\in nat \), \( n\neq 0 \)

shows \( pred(n) = n -_\mathbb{N} 1 \)proof

from assms obtain \( k \) where \( n=succ(k) \) using Nat_ZF_1_L3

with assms show \( pred(n) = n -_\mathbb{N} 1 \) using pred_succ_eq, eq_succ_imp_eq_m1

qed

For natural numbers taking the successor is the same as adding one.

lemma succ_add_one:

assumes \( n\in nat \)

shows \( succ(n) = n +_\mathbb{N} 1 \) and \( n +_\mathbb{N} 1 \in nat \) using assms

Adding and subtracting a natural number cancel each other.

lemma add_subctract:

assumes \( m\in nat \)

shows \( (m +_\mathbb{N} n) -_\mathbb{N} n = m \) using assms, diff_add_inverse2

Intervals

In this section we consider intervals of natural numbers i.e. sets of the form \(\{n+j : j \in 0..k-1\}\).

The interval is determined by two parameters: starting point and length.

Definition

\( Nat \text{Interval}(n,k) \equiv \{n +_\mathbb{N} j.\ j\in k\} \)

Subtracting the beginning af the interval results in a number from the length of the interval. It may sound weird, but note that the length of such interval is a natural number, hence a set.

lemma inter_diff_in_len:

assumes A1: \( k \in nat \) and A2: \( i \in Nat \text{Interval}(n,k) \)

shows \( i -_\mathbb{N} n \in k \)proof

from A2 obtain \( j \) where I: \( i = n +_\mathbb{N} j \) and II: \( j \in k \) using NatInterval_def

from A1, II have \( j \in nat \) using elem_nat_is_nat

moreover

from I have \( i -_\mathbb{N} n = \text{natify}(j) \) using diff_add_inverse

ultimately have \( i -_\mathbb{N} n = j \)

with II show \( thesis \)

qed

Intervals don't overlap with their starting point and the union of an interval with its starting point is the sum of the starting point and the length of the interval.

lemma length_start_decomp:

assumes A1: \( n \in nat \), \( k \in nat \)

shows \( n \cap Nat \text{Interval}(n,k) = 0 \), \( n \cup Nat \text{Interval}(n,k) = n +_\mathbb{N} k \)proof

{

fix \( i \)

assume A2: \( i \in n \) and \( i \in Nat \text{Interval}(n,k) \)

then obtain \( j \) where I: \( i = n +_\mathbb{N} j \) and II: \( j \in k \) using NatInterval_def

from A1 have \( k \in nat \) using elem_nat_is_nat

with II have \( j \in nat \) using elem_nat_is_nat

with A1, I have \( n \leq i \) using add_nat_le

moreover

from A1, A2 have \( i \lt n \) using elem_nat_is_nat

ultimately have \( False \) using le_imp_not_lt

}

thus \( n \cap Nat \text{Interval}(n,k) = 0 \)

from A1 have \( n \subseteq n +_\mathbb{N} k \) using add_nat_le

moreover {

fix \( i \)

assume \( i \in Nat \text{Interval}(n,k) \)

then obtain \( j \) where III: \( i = n +_\mathbb{N} j \) and IV: \( j \in k \) using NatInterval_def

with A1 have \( j \lt k \) using elem_nat_is_nat

with A1, III have \( i \in n +_\mathbb{N} k \) using add_lt_mono2, ltD

} ultimately have \( n \cup Nat \text{Interval}(n,k) \subseteq n +_\mathbb{N} k \)

moreover

from A1 have \( n +_\mathbb{N} k \subseteq n \cup Nat \text{Interval}(n,k) \) using nat_sum_decomp, NatInterval_def

ultimately show \( n \cup Nat \text{Interval}(n,k) = n +_\mathbb{N} k \)

qed

Some properties of three adjacent intervals.

lemma adjacent_intervals3:

assumes \( n \in nat \), \( k \in nat \), \( m \in nat \)

shows \( n +_\mathbb{N} k +_\mathbb{N} m = (n +_\mathbb{N} k) \cup Nat \text{Interval}(n +_\mathbb{N} k,m) \), \( n +_\mathbb{N} k +_\mathbb{N} m = n \cup Nat \text{Interval}(n,k +_\mathbb{N} m) \), \( n +_\mathbb{N} k +_\mathbb{N} m = n \cup Nat \text{Interval}(n,k) \cup Nat \text{Interval}(n +_\mathbb{N} k,m) \) using assms, add_assoc, length_start_decomp

end

theorem ind_on_nat:

assumes \( n\in nat \) and \( P(0) \) and \( \forall k\in nat.\ P(k)\longrightarrow P(succ(k)) \)

shows \( P(n) \)

lemma Nat_ZF_1_L3:

assumes \( n \in nat \) and \( n\neq 0 \)

shows \( \exists k\in nat.\ n = succ(k) \)

lemma empty_in_every_succ:

assumes \( n \in nat \)

shows \( 0 \in succ(n) \)

lemma nat_union_succ: shows \( nat = (\bigcup n \in nat.\ succ(n)) \)

lemma elem_nat_is_nat: