IsarMathLib

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

theory MetricSpace_ZF imports Topology_ZF_1 OrderedLoop_ZF Lattice_ZF UniformSpace_ZF

begin

A metric space is a set on which a distance between points is defined as a function \(d:X \times X \to [0,\infty)\). With this definition each metric space is a topological space which is paracompact and Hausdorff (\(T_2\)), hence normal (in fact even perfectly normal).

Pseudometric - definition and basic properties

A metric on \(X\) is usually defined as a function \(d:X \times X \to [0,\infty)\) that satisfies the conditions \(d(x,x) = 0\), \(d(x, y) = 0 \Rightarrow x = y\) (identity of indiscernibles), \(d(x, y) = d(y, x)\) (symmetry) and \(d(x, y) \le d(x, z) + d(z, y)\) (triangle inequality) for all \(x,y \in X\). Here we are going to be a bit more general and define metric and pseudo-metric as a function valued in an ordered loop.

First we define a pseudo-metric, which has the axioms of a metric, but without the second part of the identity of indiscernibles. In our definition IsApseudoMetric is a predicate on five sets: the function \(d\), the set \(X\) on which the metric is defined, the loop carrier \(G\), the loop operation \(A\) and the order \(r\) on \(G\).

definition

\( \text{IsApseudoMetric}(d,X,G,A,r) \equiv d:X\times X \rightarrow \text{Nonnegative}(G,A,r) \) \( \wedge (\forall x\in X.\ d\langle x,x\rangle = \text{ TheNeutralElement}(G,A))\) \( \wedge (\forall x\in X.\ \forall y\in X.\ d\langle x,y\rangle = d\langle y,x\rangle )\) \( \wedge (\forall x\in X.\ \forall y\in X.\ \forall z\in X.\ \langle d\langle x,z\rangle , A\langle d\langle x,y\rangle ,d\langle y,z\rangle \rangle \rangle \in r) \)

We add the full axiom of identity of indiscernibles to the definition of a pseudometric to get the definition of metric.

definition

\( \text{IsAmetric}(d,X,G,A,r) \equiv \) \( \text{IsApseudoMetric}(d,X,G,A,r) \wedge (\forall x\in X.\ \forall y\in X.\ d\langle x,y\rangle = \text{ TheNeutralElement}(G,A) \longrightarrow x=y) \)

A disk is defined as set of points located less than the radius from the center.

definition

\( \text{Disk}(X,d,r,c,R) \equiv \{x\in X.\ \langle d\langle c,x\rangle ,R\rangle \in \text{StrictVersion}(r)\} \)

We define metric topology as consisting of unions of open disks. Note the same definition is used for the topology generated by a pseudometric.

definition

\( \text{MetricTopology}(X,L,A,r,d) \equiv \{\bigcup \mathcal{A} .\ \mathcal{A} \in Pow(\bigcup c\in X.\ \{ \text{Disk}(X,d,r,c,R).\ R\in \text{PositiveSet}(L,A,r)\})\} \)

Next we define notation for metric spaces. We will reuse the additive notation defined in the loop1 locale adding only the assumption about \(d\) being a pseudometric and notation for a disk centered at \(c\) with radius \(R\). Since for many theorems it is sufficient to assume the pseudometric axioms we will assume in this context that the sets \(d,X,L,A,r\) form a pseudometric raher than a metric. In the pmetric_space context \(\tau\) denotes the topology defined by the metric \(d\). Analogously to the notation defined in the topology0 context \( int(A) \), \( cl(A) \), \( \partial A \) will denote the interior, closure and boundary of the set \(A\) with respect to the metric topology.

locale pmetric_space = loop1 +

assumes pmetricAssum: \( \text{IsApseudoMetric}(d,X,L,A,r) \)

defines \( disk(c,R) \equiv \text{Disk}(X,d,r,c,R) \)

defines \( \tau \equiv \text{MetricTopology}(X,L,A,r,d) \)

defines \( int(D) \equiv \text{Interior}(D,\tau ) \)

defines \( cl(D) \equiv \text{Closure}(D,\tau ) \)

The next lemma shows the definition of the pseudometric in the notation used in the pmetric_space context.

lemma (in pmetric_space) pmetric_properties:

shows \( d: X\times X \rightarrow L^+ \), \( \forall x\in X.\ d\langle x,x\rangle = 0 \), \( \forall x\in X.\ \forall y\in X.\ d\langle x,y\rangle = d\langle y,x\rangle \), \( \forall x\in X.\ \forall y\in X.\ \forall z\in X.\ d\langle x,z\rangle \leq d\langle x,y\rangle + d\langle y,z\rangle \) using pmetricAssum unfolding IsApseudoMetric_def

The values of the metric are in the in the nonnegative set of the loop, hence in the loop.

lemma (in pmetric_space) pmetric_loop_valued:

assumes \( x\in X \), \( y\in X \)

shows \( d\langle x,y\rangle \in L^+ \), \( d\langle x,y\rangle \in L \)proof

from assms show \( d\langle x,y\rangle \in L^+ \) using pmetric_properties(1), apply_funtype

then show \( d\langle x,y\rangle \in L \) using Nonnegative_def

qed

The definition of the disk in the notation used in the pmetric_space context:

lemma (in pmetric_space) disk_definition:

shows \( disk(c,R) = \{x\in X.\ d\langle c,x\rangle \lt R\} \)proof

have \( disk(c,R) = \text{Disk}(X,d,r,c,R) \)

then have \( disk(c,R) = \{x\in X.\ \langle d\langle c,x\rangle ,R\rangle \in \text{StrictVersion}(r)\} \) unfolding Disk_def

moreover

have \( \forall x\in X.\ \langle d\langle c,x\rangle ,R\rangle \in \text{StrictVersion}(r) \longleftrightarrow d\langle c,x\rangle \lt R \) using def_of_strict_ver

ultimately show \( thesis \)

qed

If the radius is positive then the center is in disk.

lemma (in pmetric_space) center_in_disk:

assumes \( c\in X \) and \( R\in L_+ \)

shows \( c \in disk(c,R) \) using pmetricAssum, assms, IsApseudoMetric_def, PositiveSet_def, disk_definition

A technical lemma that allows us to shorten some proofs: if \(c\) is an element of \(X\) and \(x\) is in disk with center \(c\) and radius \(R\) then \(R\) is a positive element of \(L\) and \(-d(x,y)+R\) is in the set of positive elements of the loop.

lemma (in pmetric_space) radius_in_loop:

assumes \( c\in X \) and \( x \in disk(c,R) \)

shows \( R\in L \), \( 0 \lt R \), \( R\in L_+ \), \( ( - d\langle c,x\rangle + R) \in L_+ \)proof

from assms(2) have \( x\in X \) and \( d\langle c,x\rangle \lt R \) using disk_definition

with assms(1) show \( 0 \lt R \) using pmetric_properties(1), apply_funtype, nonneg_definition, loop_strict_ord_trans

then show \( R\in L \) and \( R\in L_+ \) using posset_definition, PositiveSet_def

from \( d\langle c,x\rangle \lt R \) show \( ( - d\langle c,x\rangle + R) \in L_+ \) using ls_other_side(2)

qed

If a point \(x\) is inside a disk \(B\) and \(m\leq -d\langle c,x\rangle + R\) then the disk centered at the point \(x\) and with radius \(m\) is contained in the disk \(B\).

lemma (in pmetric_space) disk_in_disk:

assumes \( c\in X \) and \( x \in disk(c,R) \) and \( m \leq ( - d\langle c,x\rangle + R) \)

shows \( disk(x,m) \subseteq disk(c,R) \)proof

fix \( y \)

assume \( y \in disk(x,m) \)

then have \( d\langle x,y\rangle \lt m \) using disk_definition

from assms(1,2), \( y \in disk(x,m) \) have \( R\in L \), \( x\in X \), \( y\in X \) using radius_in_loop(1), disk_definition

with assms(1) have \( d\langle c,y\rangle \leq d\langle c,x\rangle + d\langle x,y\rangle \) using pmetric_properties(4)

from assms(1), \( x\in X \) have \( d\langle c,x\rangle \in L \) using pmetric_properties(1), apply_funtype, nonneg_subset

with \( d\langle x,y\rangle \lt m \), assms(3) have \( d\langle c,x\rangle + d\langle x,y\rangle \lt d\langle c,x\rangle + ( - d\langle c,x\rangle + R) \) using loop_strict_ord_trans1, strict_ord_trans_inv(2)

with \( d\langle c,x\rangle \in L \), \( R\in L \), \( d\langle c,y\rangle \leq d\langle c,x\rangle + d\langle x,y\rangle \), \( y\in X \) show \( y \in disk(c,R) \) using lrdiv_props(6), loop_strict_ord_trans, disk_definition

qed

A special case of disk_in_disk where we set \(m = -d\langle c,x\rangle + R\): if \(x\) is an element of a disk with center \(c\in X\) and radius \(R\) then this disk contains the disk centered at \(x\) and with radius \(-d\langle c,x\rangle + R\).

lemma (in pmetric_space) disk_in_disk1:

assumes \( c\in X \) and \( x \in disk(c,R) \)

shows \( disk(x, - d\langle c,x\rangle + R) \subseteq disk(c,R) \)proof

from assms(2) have \( R\in L \) and \( d\langle c,x\rangle \in L \) using disk_definition, less_members

with assms show \( thesis \) using left_right_sub_closed(1), loop_ord_refl, disk_in_disk

qed

Assuming that two disks have the same center, closed disk with smaller radius in contained in the (open) disk with a larger radius.

lemma (in pmetric_space) disk_radius_strict_mono:

assumes \( r_1 \lt r_2 \)

shows \( \{y\in X.\ d\langle x,y\rangle \leq r_1\} \subseteq disk(x,r_2) \) using assms, loop_strict_ord_trans, disk_definition

If we assume that the loop's order relation down-directs \(L_+\) then the collection of disks centered at points of the space and with radii in the positive set of the loop satisfies the base condition. The property that an order relation "down-directs" a set is defined in Order_ZF and means that every two-element subset of the set has a lower bound in that set.

lemma (in pmetric_space) disks_form_base:

assumes \( r \text{ down-directs } L_+ \)

defines \( B \equiv \bigcup c\in X.\ \{disk(c,R).\ R\in L_+\} \)

shows \( B \text{ satisfies the base condition } \)proof

{

fix \( U \) \( V \)

assume \( U\in B \), \( V\in B \)

fix \( x \)

assume \( x\in U\cap V \)

have \( \exists W\in B.\ x\in W \wedge W\subseteq U\cap V \)proof

from assms(2), \( U\in B \), \( V\in B \) obtain \( c_U \) \( c_V \) \( R_U \) \( R_V \) where \( c_U \in X \), \( R_U \in L_+ \), \( c_V \in X \), \( R_V \in L_+ \), \( U = disk(c_U,R_U) \), \( V = disk(c_V,R_V) \)

with \( x\in U\cap V \) have \( x \in disk(c_U,R_U) \) and \( x \in disk(c_V,R_V) \)

then have \( x\in X \), \( d\langle c_U,x\rangle \lt R_U \), \( d\langle c_V,x\rangle \lt R_V \) using disk_definition

let \( m_U = - d\langle c_U,x\rangle + R_U \)

let \( m_V = - d\langle c_V,x\rangle + R_V \)

from \( c_U\in X \), \( x\in disk(c_U,R_U) \), \( c_V\in X \), \( x\in disk(c_V,R_V) \) have \( m_U\in L_+ \) and \( m_V\in L_+ \) using radius_in_loop(4)

with assms(1) obtain \( m \) where \( m \in L_+ \), \( m \leq m_U \), \( m \leq m_V \) unfolding DownDirects_def

let \( W = disk(x,m) \)

from \( m \in L_+ \), \( m \leq m_U \), \( m \leq m_V \), \( c_U \in X \), \( x \in disk(c_U,R_U) \), \( c_V \in X \), \( x \in disk(c_V,R_V) \), \( U = disk(c_U,R_U) \), \( V = disk(c_V,R_V) \) have \( W \subseteq U\cap V \) using disk_in_disk

moreover

from assms(2), \( x\in X \), \( m \in L_+ \) have \( W \in B \) and \( x\in W \) using center_in_disk

ultimately show \( thesis \)

qed

}

then show \( thesis \) unfolding SatisfiesBaseCondition_def

qed

Disks centered at points farther away than the sum of radii do not overlap.

lemma (in pmetric_space) far_disks:

assumes \( x\in X \), \( y\in X \), \( r_x + r_y \leq d\langle x,y\rangle \)

shows \( disk(x,r_x)\cap disk(y,r_y) = \emptyset \)proof

{

assume \( disk(x,r_x)\cap disk(y,r_y) \neq \emptyset \)

then obtain \( z \) where \( z \in disk(x,r_x)\cap disk(y,r_y) \)

then have \( z\in X \) and \( d\langle x,z\rangle + d\langle y,z\rangle \lt r_x + r_y \) using disk_definition, add_ineq_strict

moreover

from assms(1,2), \( z\in X \) have \( d\langle x,y\rangle \leq d\langle x,z\rangle + d\langle y,z\rangle \) using pmetric_properties(3,4)

ultimately have \( d\langle x,y\rangle \lt r_x + r_y \) using loop_strict_ord_trans

with assms(3) have \( False \) using loop_strict_ord_trans

}

thus \( thesis \)

qed

If we have a loop element that is smaller than the distance between two points, then we can separate these points with disks.

lemma (in pmetric_space) disjoint_disks:

assumes \( x\in X \), \( y\in X \), \( r_x \lt d\langle x,y\rangle \)

shows \( ( - r_x + (d\langle x,y\rangle )) \in L_+ \) and \( disk(x,r_x)\cap disk(y, - r_x + (d\langle x,y\rangle )) = 0 \)proof

from assms(3) show \( ( - r_x + (d\langle x,y\rangle )) \in L_+ \) using ls_other_side, posset_definition1

from assms(1,2,3) have \( r_x\in L \) and \( d\langle x,y\rangle \in L \) using less_members(1), pmetric_loop_valued(2)

then have \( r_x + ( - r_x + (d\langle x,y\rangle )) = d\langle x,y\rangle \) using lrdiv_props(6)

with assms(1,2), \( d\langle x,y\rangle \in L \) show \( disk(x,r_x)\cap disk(y, - r_x + (d\langle x,y\rangle )) = 0 \) using loop_ord_refl, far_disks

qed

The definition of metric topology written in notation of pmetric_space context:

lemma (in pmetric_space) metric_top_def_alt:

defines \( B \equiv \bigcup c\in X.\ \{disk(c,R).\ R\in L_+\} \)

shows \( \tau = \{\bigcup A.\ A \in Pow(B)\} \)proof

from assms have \( \text{MetricTopology}(X,L,A,r,d) = \{\bigcup A.\ A \in Pow(B)\} \) unfolding MetricTopology_def

thus \( thesis \)

qed

If the order of the loop down-directs its set of positive elements then the metric topology defined as collection of unions of (open) disks is indeed a topology. Recall that in the pmetric_space context \(\tau\) denotes the metric topology.

theorem (in pmetric_space) pmetric_is_top:

assumes \( r \text{ down-directs } L_+ \)

shows \( \tau \text{ is a topology } \) using assms, disks_form_base, Top_1_2_T1, metric_top_def_alt

If \(r\) down-directs \(L_+\) then the collection of open disks is a base for the metric topology.

theorem (in pmetric_space) disks_are_base:

assumes \( r \text{ down-directs } L_+ \)

defines \( B \equiv \bigcup c\in X.\ \{disk(c,R).\ R\in L_+\} \)

shows \( B \text{ is a base for } \tau \) using assms, disks_form_base, Top_1_2_T1, metric_top_def_alt

If \(r\) down-directs \(L_+\) then \(X\) is the carrier of metric topology.

theorem (in pmetric_space) metric_top_carrier:

assumes \( r \text{ down-directs } L_+ \)

shows \( \bigcup \tau = X \)proof

let \( B = \bigcup c\in X.\ \{disk(c,R).\ R\in L_+\} \)

from assms have \( \bigcup \tau = \bigcup B \) using disks_are_base, Top_1_2_L5

moreover

have \( \bigcup B = X \)proof

show \( \bigcup B \subseteq X \) using disk_definition

from assms show \( X \subseteq \bigcup B \) unfolding DownDirects_def using center_in_disk

qed

ultimately show \( \bigcup \tau = X \)

qed

Under the assumption that \(r\) down-directs \(L_+\) the propositions proven in the topology0 context can be used in the pmetric_space context.

lemma (in pmetric_space) topology0_valid_in_pmetric_space:

assumes \( r \text{ down-directs } L_+ \)

shows \( topology0(\tau ) \) using assms, pmetric_is_top unfolding topology0_def

If \(r\) down-directs \(L_+\) then disks are open in the metric topology.

lemma (in pmetric_space) disks_open:

assumes \( c\in X \), \( R\in L_+ \), \( r \text{ down-directs } L_+ \)

shows \( disk(c,R) \in \tau \) using assms, base_sets_open, disks_are_base(1), pmetric_is_top

If \(r\) down-directs \(L_+\) and \(x\) is an element of an open set \(U\) then there exist radius \(R\in L_+\) such that the disk with center \(x\) and radius \(R\) is contained in \(U\).

lemma (in pmetric_space) point_open_disk:

assumes \( r \text{ down-directs } L_+ \), \( U\in \tau \), \( x\in U \)

shows \( \exists R\in L_+.\ disk(x,R) \subseteq U \)proof

let \( B = \bigcup c\in X.\ \{disk(c,R).\ R\in L_+\} \)

from assms have \( \exists V\in B.\ V\subseteq U \wedge x\in V \) using disks_are_base, point_open_base_neigh

then obtain \( c \) \( R \) where \( c\in X \), \( R\in L_+ \), \( x\in disk(c,R) \), \( disk(c,R)\subseteq U \)

then show \( thesis \) using radius_in_loop(4), disk_in_disk1

qed

If \(r\) down-directs \(L_+\) then the generated topology cannot distinguish two points if their distance is zero. "Cannot distinguish" here means that if one is in an open set then the second one is in that set too.

lemma (in pmetric_space) zero_dist_same_open:

assumes \( r \text{ down-directs } L_+ \), \( U\in \tau \), \( x\in U \), \( y\in X \), \( d\langle x,y\rangle = 0 \)

shows \( y\in U \) using assms, point_open_disk, posset_definition1, disk_definition

A pseudometric that induces a \(T_0\) topology is a metric.

theorem (in pmetric_space) pmetric_t0_metric:

assumes \( r \text{ down-directs } L_+ \) and \( \tau \text{ is }T_0 \)

shows \( \text{IsAmetric}(d,X,L,A,r) \)proof

{

fix \( x \) \( y \)

assume \( x\in X \), \( y\in X \), \( d\langle x,y\rangle = 0 \)

with assms(1) have \( \forall U\in \tau .\ (x\in U \longleftrightarrow y\in U) \) using zero_dist_same_open, pmetric_properties(3)

with assms, \( x\in X \), \( y\in X \) have \( x=y \) using metric_top_carrier, Top_1_1_L1

}

then show \( \text{IsAmetric}(d,X,L,A,r) \) using pmetricAssum unfolding IsAmetric_def

qed

To define the metric_space locale we take the pmetric_space and add the assumption of identity of indiscernibles.

locale metric_space = pmetric_space +

assumes ident_indisc: \( \forall x\in X.\ \forall y\in X.\ d\langle x,y\rangle = 0 \longrightarrow x=y \)

In the metric_space locale \(d\) is a metric.

lemma (in metric_space) d_metric:

shows \( \text{IsAmetric}(d,X,L,A,r) \) using pmetricAssum, ident_indisc unfolding IsAmetric_def

Distance of different points is greater than zero.

lemma (in metric_space) dist_pos:

assumes \( x\in X \), \( y\in X \), \( x\neq y \)

shows \( 0 \lt d\langle x,y\rangle \), \( d\langle x,y\rangle \in L_+ \)proof

from assms(1,2) have \( d\langle x,y\rangle \in L^+ \) using pmetric_properties(1), apply_funtype

then have \( 0 \leq d\langle x,y\rangle \) using Nonnegative_def

with assms show \( d\langle x,y\rangle \in L_+ \) and \( 0 \lt d\langle x,y\rangle \) using ident_indisc, posset_definition, posset_definition1

qed

If \(r\) down-directs \(L_+\) then the ordered loop valued metric space is \(T_2\) (i.e. Hausdorff).

theorem (in metric_space) metric_space_T2:

assumes \( r \text{ down-directs } L_+ \)

shows \( \tau \text{ is }T_2 \)proof

let \( B = \bigcup c\in X.\ \{disk(c,R).\ R\in L_+\} \)

{

fix \( x \) \( y \)

assume \( x\in \bigcup \tau \), \( y\in \bigcup \tau \), \( x\neq y \)

from assms have \( B\subseteq \tau \) using metric_top_def_alt

have \( \exists U\in B.\ \exists V\in B.\ x\in U \wedge y\in V \wedge U\cap V = \emptyset \)proof

let \( R = d\langle x,y\rangle \)

from assms have \( \bigcup \tau = X \) using metric_top_carrier

with \( x\in \bigcup \tau \) have \( x\in X \)

from \( \bigcup \tau = X \), \( y\in \bigcup \tau \) have \( y\in X \)

with \( x\neq y \), \( x\in X \) have \( R\in L_+ \) using dist_pos

with \( x\in X \), \( y\in X \) have \( disk(x,R) \in B \) and \( disk(y,R) \in B \)

{

assume \( disk(x,R) \cap disk(y,R) = \emptyset \)

moreover

from \( x\in X \), \( y\in X \), \( R\in L_+ \) have \( disk(x,R)\in B \), \( disk(y,R)\in B \), \( x\in disk(x,R) \), \( y\in disk(y,R) \) using center_in_disk

ultimately have \( \exists U\in B.\ \exists V\in B.\ x\in U \wedge y\in V \wedge U\cap V = 0 \)

}

moreover {

assume \( disk(x,R) \cap disk(y,R) \neq 0 \)

then obtain \( z \) where \( z \in disk(x,R) \) and \( z \in disk(y,R) \)

then have \( d\langle x,z\rangle \lt R \) using disk_definition

then have \( 0 \lt - d\langle x,z\rangle + R \) using ls_other_side(1)

let \( r = - d\langle x,z\rangle + R \)

have \( r \lt R \)proof

from \( z \in disk(y,R) \), \( x\in X \), \( y\in X \) have \( z\in X \), \( x\neq z \) using disk_definition, pmetric_properties(3)

with \( x\in X \), \( y\in X \), \( z\in X \) show \( thesis \) using pmetric_loop_valued, dist_pos(1), add_subtract_pos(2)

qed

with \( x\in X \), \( y\in X \) have \( disk(x,r)\cap disk(y, - r + R) = 0 \) by (rule disjoint_disks )

moreover

from \( 0 \lt r \), \( r \lt R \) have \( r\in L_+ \), \( ( - r + R) \in L_+ \) using ls_other_side, posset_definition1

with \( x\in X \), \( y\in X \) have \( disk(x,r)\in B \), \( disk(y, - r + (d\langle x,y\rangle ))\in B \) and \( x\in disk(x,r) \), \( y\in disk(y, - r + (d\langle x,y\rangle )) \) using center_in_disk

ultimately have \( \exists U\in B.\ \exists V\in B.\ x\in U \wedge y\in V \wedge U\cap V = 0 \)

} ultimately show \( thesis \)

qed

with \( B\subseteq \tau \) have \( \exists U\in \tau .\ \exists V\in \tau .\ x\in U \wedge y\in V \wedge U\cap V = \emptyset \) by (rule exist2_subset )

}

then show \( thesis \) unfolding isT2_def

qed

Uniform structures on (pseudo-)metric spaces

Each pseudometric space with pseudometric \(d:X\times X\rightarrow L^+\) supports a natural uniform structure, defined as supersets of the collection of inverse images \(U_c = d^{-1}([0,c])\), where \(c>0\).

In the following definition \(X\) is the underlying space, \(L\) is the loop (carrier), \(A\) is the loop operation, \(r\) is an order relation compatible with \(A\), and \(d\) is a pseudometric on \(X\), valued in the ordered loop \(L\). With this we define the uniform gauge as the collection of inverse images of the closed intervals \([0,c]\) as \(c\) varies of the set of positive elements of \(L\). See uniform_gauge_def_alt for this definition in a more readable notation.

definition

\( \text{UniformGauge}(X,L,A,r,d) \equiv \{d^{-1}(\{c\in \text{Nonnegative}(L,A,r).\ \langle c,b\rangle \in r\}).\ b\in \text{PositiveSet}(L,A,r)\} \)

In the pmetric_space context we will write \( \text{UniformGauge}(X,L,A,r,d) \) as \( \mathfrak{B} \).

abbreviation (in pmetric_space)

\( \mathfrak{B} \equiv \text{UniformGauge}(X,L,A,r,d) \)

In notation defined in the pmetric_space context we can write the uniform gauge as \(\{ d^{-1}(\{c\in L^+: c\leq b\}): b \in L_+ \}\).

lemma (in pmetric_space) uniform_gauge_def_alt:

shows \( \mathfrak{B} = \{d^{-1}(\{c\in L^+.\ c\leq b\}).\ b\in L_+\} \) unfolding UniformGauge_def

Members of the uniform gauge are subsets of \(X\times X\) i.e. relations on \(X\).

lemma (in pmetric_space) uniform_gauge_relations:

assumes \( B\in \mathfrak{B} \)

shows \( B\subseteq X\times X \) using assms, uniform_gauge_def_alt, pmetric_properties(1), func1_1_L3

If the distance between two points of \(X\) is less or equal \(b\), then this pair of points is in \(d^{-1}([0,b])\).

lemma (in pmetric_space) gauge_members:

assumes \( x\in X \), \( y\in X \), \( d\langle x,y\rangle \leq b \)

shows \( \langle x,y\rangle \in d^{-1}(\{c\in L^+.\ c\leq b\}) \) using assms, pmetric_properties(1), apply_funtype, func1_1_L15

Suppose \(b\in L_+\) (i.e. b is an element of the loop that is greater than the neutral element) and \(x\in X\). Then the image of the singleton set \(\{ x\}\) by the relation \(B=\{ d^{-1}(\{c\in L^+: c\leq b\}\) is the set \(\{ y\in X:d\langle x,y\rangle \leq b\}\), i.e. the closed disk with center \(x\) and radius \(b\). Hence the the image \(B\{ x\}\) contains the open disk with center \(x\) and radius \(b\).

lemma (in pmetric_space) disk_in_gauge:

assumes \( b\in L_+ \), \( x\in X \)

defines \( B \equiv d^{-1}(\{c\in L^+.\ c\leq b\}) \)

shows \( B\{x\} = \{y\in X.\ d\langle x,y\rangle \leq b\} \) and \( disk(x,b) \subseteq B\{x\} \)proof

from assms(1,3) have \( B\subseteq X\times X \) using uniform_gauge_def_alt, uniform_gauge_relations

with assms(2,3) show \( B\{x\} = \{y\in X.\ d\langle x,y\rangle \leq b\} \) using pmetric_properties(1), func1_1_L15

then show \( disk(x,b) \subseteq B\{x\} \) using disk_definition

qed

Gauges corresponding to larger elements of the loop are larger.

lemma (in pmetric_space) uniform_gauge_mono:

assumes \( b_1\leq b_2 \)

shows \( d^{-1}(\{c\in L^+.\ c\leq b_1\}) \subseteq d^{-1}(\{c\in L^+.\ c\leq b_2\}) \) using ordLoopAssum, assms, vimage_mono1 unfolding IsAnOrdLoop_def, IsPartOrder_def, trans_def

For any two sets of the form \(d^{-1}([0,b])\) we can find a third one that is contained in both.

lemma (in pmetric_space) gauge_1st_cond:

assumes \( r \text{ down-directs } L_+ \), \( B_1\in \mathfrak{B} \), \( B_2\in \mathfrak{B} \)

shows \( \exists B_3\in \mathfrak{B} .\ B_3\subseteq B_1\cap B_2 \)proof

from assms(2,3) obtain \( b_1 \) \( b_2 \) where \( b_1\in L_+ \), \( b_2\in L_+ \) and I: \( B_1 = d^{-1}(\{c\in L^+.\ c\leq b_1\}) \), \( B_2 = d^{-1}(\{c\in L^+.\ c\leq b_2\}) \) using uniform_gauge_def_alt

from assms(1), \( b_1\in L_+ \), \( b_2\in L_+ \) obtain \( b_3 \) where \( b_3\in L_+ \), \( b_3\leq b_1 \), \( b_3\leq b_2 \) unfolding DownDirects_def

from I, \( b_3\leq b_1 \), \( b_3\leq b_2 \) have \( d^{-1}(\{c\in L^+.\ c\leq b_3\}) \subseteq B_1\cap B_2 \) using uniform_gauge_mono

with \( b_3\in L_+ \) show \( thesis \) using uniform_gauge_def_alt

qed

Sets of the form \(d^{-1}([0,b])\) contain the diagonal.

lemma (in pmetric_space) gauge_2nd_cond:

assumes \( B\in \mathfrak{B} \)

shows \( id(X)\subseteq B \)proof

fix \( p \)

assume \( p\in id(X) \)

then obtain \( x \) where \( x\in X \) and \( p=\langle x,x\rangle \)

then have \( p\in X\times X \) and \( d(p) = 0 \) using pmetric_properties(2)

from assms obtain \( b \) where \( b\in L_+ \) and \( B = d^{-1}(\{c\in L^+.\ c\leq b\}) \) using uniform_gauge_def_alt

with \( p\in X\times X \), \( d(p) = 0 \) show \( p\in B \) using posset_definition1, loop_zero_nonneg, pmetric_properties(1), func1_1_L15

qed

Sets of the form \(d^{-1}([0,b])\) are symmetric.

lemma (in pmetric_space) gauge_symmetric:

assumes \( B\in \mathfrak{B} \)

shows \( B = converse(B) \)proof

from assms obtain \( b \) where \( B = d^{-1}(\{c\in L^+.\ c\leq b\}) \) using uniform_gauge_def_alt

with pmetricAssum show \( thesis \) unfolding IsApseudoMetric_def using symm_vimage_symm

qed

A set of the form \(d^{-1}([0,b])\) contains a symmetric set of this form.

corollary (in pmetric_space) gauge_3rd_cond:

assumes \( B_1\in \mathfrak{B} \)

shows \( \exists B_2\in \mathfrak{B} .\ B_2 \subseteq converse(B_1) \) using assms, gauge_symmetric

The collection of sets of the form \(d^{-1}([0,b])\) for \(b\in L_+\) is contained of the powerset of \(X\times X\).

lemma (in pmetric_space) gauge_5thCond:

shows \( \mathfrak{B} \subseteq Pow(X\times X) \) using uniform_gauge_def_alt, pmetric_properties(1), func1_1_L3

If the set of positive values is non-empty, then there are sets of the form \(d^{-1}([0,b])\) for \(b>0\).

lemma (in pmetric_space) gauge_6thCond:

assumes \( L_+\neq \emptyset \)

shows \( \mathfrak{B} \neq \emptyset \) using assms, uniform_gauge_def_alt

The remaining 4th condition for the sets of the form \(d^{-1}([0,b])\) to be a uniform base (or a fundamental system of entourages) cannot be proven without additional assumptions in the context of ordered loop valued metrics. To see that consider the example of natural numbers with the metric \(d\langle x,y \rangle = |x-y|\), where we think of \(d\) as valued in the nonnegative set of ordered group of integers. Now take the set \(B_1 = d^{-1}([0,1]) = d^{-1}(\{ 0,1\} )\). Then the set \(B_1 \circ B_1\) is strictly larger than \(B_1\), but there is no smaller set \(B_2\) we can take so that \(B_2 \circ B_2 \subseteq B_1\). One condition that is sufficient is that for every \(b_1 >0\) there is a \(b_2 >0\) such that \(b_2 + b_2 \leq b_1 \). I have not found a standard name for this property, for now we will use the name IsHalfable.

definition

\( \text{IsHalfable}(L,A,r) \equiv \forall b_1\in \text{PositiveSet}(L,A,r).\ \exists b_2\in \text{PositiveSet}(L,A,r).\ \langle A\langle b_2,b_2\rangle ,b_1\rangle \in r \)

The property of halfability written in the notation used in the pmetric_space context.

lemma (in pmetric_space) is_halfable_def_alt:

assumes \( \text{IsHalfable}(L,A,r) \), \( b_1\in L_+ \)

shows \( \exists b_2\in L_+.\ b_2 + b_2 \leq b_1 \) using assms unfolding IsHalfable_def

If \(B_i = d^{-1}(\{c \in L_+: c\leq b_i\})\) for \(i=1,2\) and \(b_2+b_2\leq b1\) then \(B_2\circ B_2 \leq B_1\). The proof uses the triangle inequality so it's not really a property of ordered loops only.

lemma (in pmetric_space) half_vimage_square:

assumes \( b_2\in L_+ \) and \( b_2 + b_2 \leq b_1 \)

defines \( B_1 \equiv d^{-1}(\{c\in L^+.\ c\leq b_1\}) \) and \( B_2 \equiv d^{-1}(\{c\in L^+.\ c\leq b_2\}) \)

shows \( B_2\circ B_2 \subseteq B_1 \)proof

from assms(1,4) have \( B_2\subseteq X\times X \) using pmetric_properties(1), func1_1_L3

fix \( p \)

assume \( p \in B_2\circ B_2 \)

with \( B_2\subseteq X\times X \) obtain \( x \) \( y \) where \( x\in X \), \( y\in X \) and \( p=\langle x,y\rangle \)

from \( p \in B_2\circ B_2 \), \( p=\langle x,y\rangle \) obtain \( z \) where \( \langle x,z\rangle \in B_2 \) and \( \langle z,y\rangle \in B_2 \) using rel_compdef

with \( B_2\subseteq X\times X \) have \( z\in X \)

from assms(4), \( \langle x,z\rangle \in B_2 \), \( \langle z,y\rangle \in B_2 \) have \( d\langle x,z\rangle + d\langle z,y\rangle \leq b_2 + b_2 \) using pmetric_properties(1), func1_1_L15, add_ineq

with \( b_2 + b_2 \leq b_1 \) have \( d\langle x,z\rangle + d\langle z,y\rangle \leq b_1 \) using loop_ord_trans

with assms(3), \( x\in X \), \( y\in X \), \( z\in X \), \( p=\langle x,y\rangle \) show \( p\in B_1 \) using pmetric_properties(4), loop_ord_trans, gauge_members

qed

If the loop order is halfable then for every set \(B_1\) of the form \(d^{-1}([0,b_1])\) for some \(b_1>0\) we can find another one \(B_2 = d^{-1}([0,b_2])\) such that \(B_2\) composed with itself is contained in \(B_1\).

lemma (in pmetric_space) gauge_4thCond:

assumes \( \text{IsHalfable}(L,A,r) \), \( B_1\in \mathfrak{B} \)

shows \( \exists B_2\in \mathfrak{B} .\ B_2\circ B_2 \subseteq B_1 \)proof

from assms(2) obtain \( b_1 \) where \( b_1\in L_+ \) and \( B_1 = d^{-1}(\{c\in L^+.\ c\leq b_1\}) \) using uniform_gauge_def_alt

from assms(1), \( b_1\in L_+ \) obtain \( b_2 \) where \( b_2\in L_+ \) and \( b_2 + b_2 \leq b_1 \) using is_halfable_def_alt

with \( B_1 = d^{-1}(\{c\in L^+.\ c\leq b_1\}) \) show \( thesis \) using half_vimage_square unfolding UniformGauge_def

qed

If \(X\) and \(L_+\) are not empty, the order relation \(r\) down-directs \(L_+\), and the loop order is halfable, then \(\mathfrak{B}\) (which in the pmetric_space context is an abbreviation for \(\{ d^{-1}(\{c\in L^+: c\leq b\}: b \in L_+ \}\)) is a fundamental system of entourages, hence its supersets form a uniformity on \(X\) and hence those supersets define a topology on \(X\).

theorem (in pmetric_space) metric_gauge_base:

assumes \( X\neq \emptyset \), \( L_+\neq \emptyset \), \( r \text{ down-directs } L_+ \), \( \text{IsHalfable}(L,A,r) \)

shows \( \mathfrak{B} \text{ is a uniform base on } X \), \( \text{Supersets}(X\times X,\mathfrak{B} ) \text{ is a uniformity on } X \), \( \text{UniformTopology}( \text{Supersets}(X\times X,\mathfrak{B} ),X) \text{ is a topology } \), \( \bigcup \text{UniformTopology}( \text{Supersets}(X\times X,\mathfrak{B} ),X) = X \) using assms, gauge_1st_cond, gauge_2nd_cond, gauge_3rd_cond, gauge_4thCond, gauge_5thCond, gauge_6thCond, uniformity_base_is_base, uniform_top_is_top unfolding IsUniformityBaseOn_def

At this point we know that a pseudometric induces two topologies: one consisting of unions of open disks (the metric topology) and second one being the uniform topology derived from the uniformity generated the fundamental system of entourages (the base uniformity) of the sets of the form \(d^{-1}([0,b])\) for \(b>0\). The next theorem states that if \(X\) and \(L_+\) are not empty, \(r\) down-directs \(L_+\), and the loop order is halfable, then these two topologies are in fact the same. Recall that in the pmetric_space context \(\tau\) denotes the metric topology.

theorem (in pmetric_space) metric_top_is_uniform_top:

assumes \( X\neq \emptyset \), \( L_+\neq \emptyset \), \( r \text{ down-directs } L_+ \), \( \text{IsHalfable}(L,A,r) \)

shows \( \tau = \text{UniformTopology}( \text{Supersets}(X\times X,\mathfrak{B} ),X) \)proof

let \( \Phi = \text{Supersets}(X\times X,\mathfrak{B} ) \)

from assms have \( \Phi \text{ is a uniformity on } X \) using metric_gauge_base

let \( T = \text{UniformTopology}(\Phi ,X) \)

{

fix \( U \)

assume \( U\in T \)

then have \( U\in Pow(X) \) and I: \( \forall x\in U.\ U\in \{V\{x\}.\ V\in \Phi \} \) unfolding UniformTopology_def

{

fix \( x \)

assume \( x\in U \)

with I obtain \( A \) where \( A\in \Phi \) and \( U = A\{x\} \)

from \( x\in U \), \( U\in T \) have \( x\in \bigcup T \)

with assms have \( x\in X \) using metric_gauge_base(4)

from \( A\in \Phi \) obtain \( B \) where \( B\in \mathfrak{B} \) and \( B\subseteq A \) unfolding Supersets_def

from \( B\in \mathfrak{B} \) obtain \( b \) where \( b\in L_+ \) and \( B = d^{-1}(\{c\in L^+.\ c\leq b\}) \) using uniform_gauge_def_alt

with \( x\in X \), \( B\subseteq A \), \( U = A\{x\} \) have \( disk(x,b) \subseteq U \) using disk_in_gauge(2)

with assms(3), \( x\in X \), \( b\in L_+ \) have \( \exists V\in \tau .\ x\in V \wedge V\subseteq U \) using disks_open, center_in_disk

}

with assms(3) have \( U\in \tau \) using topology0_valid_in_pmetric_space, open_neigh_open

}

thus \( T \subseteq \tau \)

let \( \mathcal{D} = \bigcup c\in X.\ \{disk(c,R).\ R\in L_+\} \)

{

fix \( U \)

assume \( U \in \mathcal{D} \)

then obtain \( c \) \( b \) where \( c\in X \), \( b\in L_+ \), \( U = disk(c,b) \)

{

fix \( x \)

assume \( x\in U \)

let \( b_1 = - d\langle c,x\rangle + b \)

from \( x\in U \), \( c\in X \), \( U = disk(c,b) \) have \( x\in X \), \( x\in disk(c,b) \), \( disk(x,b_1) \subseteq U \), \( b_1 \in L_+ \) using disk_in_disk1, disk_definition, radius_in_loop(4)

with assms(4) obtain \( b_2 \) where \( b_2\in L_+ \) and \( b_2 + b_2 \leq b_1 \) using is_halfable_def_alt

let \( D = \{y\in X.\ d\langle x,y\rangle \leq b_2\} \)

from \( b_2\in L_+ \), \( b_2 + b_2 \leq b_1 \) have \( D \subseteq disk(x,b_1) \) using posset_definition1, positive_subset, add_subtract_pos(3), loop_strict_ord_trans1, disk_radius_strict_mono

let \( B = d^{-1}(\{c\in L^+.\ c\leq b_2\}) \)

from \( b_2\in L_+ \) have \( B\in \mathfrak{B} \) using uniform_gauge_def_alt

then have \( B\in \Phi \) using uniform_gauge_relations, superset_gen

from \( b_2\in L_+ \), \( x\in X \), \( D \subseteq disk(x,b_1) \), \( disk(x,b_1) \subseteq U \) have \( B\{x\} \subseteq U \) using disk_in_gauge(1)

with \( B\in \Phi \) have \( \exists W\in \Phi .\ W\{x\} \subseteq U \)

}

with \( U = disk(c,b) \), \( \Phi \text{ is a uniformity on } X \) have \( U \in T \) using disk_definition, uniftop_def_alt1

}

hence \( \mathcal{D} \subseteq T \)

with assms show \( \tau \subseteq T \) using disks_are_base(1), metric_gauge_base(3), base_smallest_top

qed

end

Definition of IsApseudoMetric: \( \text{IsApseudoMetric}(d,X,G,A,r) \equiv d:X\times X \rightarrow \text{Nonnegative}(G,A,r) \) \( \wedge (\forall x\in X.\ d\langle x,x\rangle = \text{ TheNeutralElement}(G,A))\) \( \wedge (\forall x\in X.\ \forall y\in X.\ d\langle x,y\rangle = d\langle y,x\rangle )\) \( \wedge (\forall x\in X.\ \forall y\in X.\ \forall z\in X.\ \langle d\langle x,z\rangle , A\langle d\langle x,y\rangle ,d\langle y,z\rangle \rangle \rangle \in r) \)

lemma (in pmetric_space) pmetric_properties: shows \( d: X\times X \rightarrow L^+ \), \( \forall x\in X.\ d\langle x,x\rangle = 0 \), \( \forall x\in X.\ \forall y\in X.\ d\langle x,y\rangle = d\langle y,x\rangle \), \( \forall x\in X.\ \forall y\in X.\ \forall z\in X.\ d\langle x,z\rangle \leq d\langle x,y\rangle + d\langle y,z\rangle \)

Definition of Nonnegative: \( \text{Nonnegative}(L,A,r) \equiv \{x\in L.\ \langle \text{ TheNeutralElement}(L,A),x\rangle \in r\} \)

Definition of Disk: \( \text{Disk}(X,d,r,c,R) \equiv \{x\in X.\ \langle d\langle c,x\rangle ,R\rangle \in \text{StrictVersion}(r)\} \)

lemma def_of_strict_ver: shows \( \langle x,y\rangle \in \text{StrictVersion}(r) \longleftrightarrow \langle x,y\rangle \in r \wedge x\neq y \)

Definition of PositiveSet: \( \text{PositiveSet}(L,A,r) \equiv \) \( \{x\in L.\ \langle \text{ TheNeutralElement}(L,A),x\rangle \in r \wedge \text{ TheNeutralElement}(L,A)\neq x\} \)

lemma (in pmetric_space) disk_definition: shows \( disk(c,R) = \{x\in X.\ d\langle c,x\rangle \lt R\} \)

lemma (in loop1) nonneg_definition: shows \( x \in L^+ \longleftrightarrow 0 \leq x \)

lemma (in loop1) loop_strict_ord_trans:

assumes \( x\leq y \) and \( y \lt z \)

shows \( x \lt z \)

lemma (in loop1) posset_definition: shows \( x \in L_+ \longleftrightarrow ( 0 \leq x \wedge x\neq 0 ) \)

lemma (in loop1) ls_other_side:

assumes \( x \lt y \)

shows \( 0 \lt - x + y \), \( ( - x + y) \in L_+ \), \( 0 \lt y - x \), \( (y - x) \in L_+ \)

lemma (in pmetric_space) radius_in_loop:

assumes \( c\in X \) and \( x \in disk(c,R) \)

shows \( R\in L \), \( 0 \lt R \), \( R\in L_+ \), \( ( - d\langle c,x\rangle + R) \in L_+ \)

lemma (in loop1) nonneg_subset: shows \( L^+ \subseteq L \)

lemma (in loop1) loop_strict_ord_trans1:

assumes \( x \lt y \) and \( y\leq z \)

shows \( x \lt z \)

lemma (in loop1) strict_ord_trans_inv:

assumes \( x \lt y \), \( z\in L \)

shows \( x + z \lt y + z \) and \( z + x \lt z + y \)

lemma (in quasigroup0) lrdiv_props:

assumes \( x\in G \), \( y\in G \)

shows \( \exists !z.\ z\in G \wedge z\cdot x = y \), \( y/ x \in G \), \( (y/ x)\cdot x = y \) and \( \exists !z.\ z\in G \wedge x\cdot z = y \), \( x\backslash y \in G \), \( x\cdot (x\backslash y) = y \)

lemma (in loop1) less_members:

assumes \( x \lt y \)

shows \( x\in L \) and \( y\in L \)

lemma (in loop1) left_right_sub_closed:

assumes \( x\in L \), \( y\in L \)

shows \( ( - x + y) \in L \) and \( x - y \in L \)

lemma (in loop1) loop_ord_refl:

assumes \( x\in L \)

shows \( x\leq x \)

lemma (in pmetric_space) disk_in_disk:

assumes \( c\in X \) and \( x \in disk(c,R) \) and \( m \leq ( - d\langle c,x\rangle + R) \)

shows \( disk(x,m) \subseteq disk(c,R) \)

lemma (in pmetric_space) radius_in_loop:

assumes \( c\in X \) and \( x \in disk(c,R) \)

shows \( R\in L \), \( 0 \lt R \), \( R\in L_+ \), \( ( - d\langle c,x\rangle + R) \in L_+ \)

Definition of DownDirects: \( r \text{ down-directs } X \equiv X\neq 0 \wedge (\forall x\in X.\ \forall y\in X.\ \exists z\in X.\ \langle z,x\rangle \in r \wedge \langle z,y\rangle \in r) \)

lemma (in pmetric_space) center_in_disk:

assumes \( c\in X \) and \( R\in L_+ \)

shows \( c \in disk(c,R) \)

Definition of SatisfiesBaseCondition: \( B \text{ satisfies the base condition } \equiv \) \( \forall U V.\ ((U\in B \wedge V\in B) \longrightarrow (\forall x \in U\cap V.\ \exists W\in B.\ x\in W \wedge W \subseteq U\cap V)) \)

lemma (in loop1) add_ineq_strict:

assumes \( x \lt y \), \( z \lt t \)

shows \( x + z \lt y + t \)

lemma (in loop1) ls_other_side:

assumes \( x \lt y \)

shows \( 0 \lt - x + y \), \( ( - x + y) \in L_+ \), \( 0 \lt y - x \), \( (y - x) \in L_+ \)

lemma (in loop1) posset_definition1: shows \( x \in L_+ \longleftrightarrow 0 \lt x \)

lemma (in loop1) less_members:

assumes \( x \lt y \)

shows \( x\in L \) and \( y\in L \)

lemma (in pmetric_space) pmetric_loop_valued:

assumes \( x\in X \), \( y\in X \)

shows \( d\langle x,y\rangle \in L^+ \), \( d\langle x,y\rangle \in L \)

lemma (in pmetric_space) far_disks:

assumes \( x\in X \), \( y\in X \), \( r_x + r_y \leq d\langle x,y\rangle \)

shows \( disk(x,r_x)\cap disk(y,r_y) = \emptyset \)

Definition of MetricTopology: \( \text{MetricTopology}(X,L,A,r,d) \equiv \{\bigcup \mathcal{A} .\ \mathcal{A} \in Pow(\bigcup c\in X.\ \{ \text{Disk}(X,d,r,c,R).\ R\in \text{PositiveSet}(L,A,r)\})\} \)

lemma (in pmetric_space) disks_form_base:

assumes \( r \text{ down-directs } L_+ \)

defines \( B \equiv \bigcup c\in X.\ \{disk(c,R).\ R\in L_+\} \)

shows \( B \text{ satisfies the base condition } \)

theorem Top_1_2_T1:

assumes \( B \text{ satisfies the base condition } \) and \( T = \{\bigcup A.\ A\in Pow(B)\} \)

shows \( T \text{ is a topology } \) and \( B \text{ is a base for } T \)

lemma (in pmetric_space) metric_top_def_alt:

defines \( B \equiv \bigcup c\in X.\ \{disk(c,R).\ R\in L_+\} \)

shows \( \tau = \{\bigcup A.\ A \in Pow(B)\} \)

theorem (in pmetric_space) disks_are_base:

assumes \( r \text{ down-directs } L_+ \)

defines \( B \equiv \bigcup c\in X.\ \{disk(c,R).\ R\in L_+\} \)

shows \( B \text{ is a base for } \tau \)

lemma Top_1_2_L5:

assumes \( B \text{ is a base for } T \)

shows \( \bigcup T = \bigcup B \)

theorem (in pmetric_space) pmetric_is_top:

assumes \( r \text{ down-directs } L_+ \)

shows \( \tau \text{ is a topology } \)

lemma base_sets_open:

assumes \( B \text{ is a base for } T \) and \( U \in B \)

shows \( U \in T \)

theorem (in pmetric_space) disks_are_base:

assumes \( r \text{ down-directs } L_+ \)

defines \( B \equiv \bigcup c\in X.\ \{disk(c,R).\ R\in L_+\} \)

shows \( B \text{ is a base for } \tau \)

lemma point_open_base_neigh:

assumes \( B \text{ is a base for } T \) and \( U\in T \) and \( x\in U \)

shows \( \exists V\in B.\ V\subseteq U \wedge x\in V \)

lemma (in pmetric_space) disk_in_disk1:

assumes \( c\in X \) and \( x \in disk(c,R) \)

shows \( disk(x, - d\langle c,x\rangle + R) \subseteq disk(c,R) \)

lemma (in pmetric_space) point_open_disk:

assumes \( r \text{ down-directs } L_+ \), \( U\in \tau \), \( x\in U \)

shows \( \exists R\in L_+.\ disk(x,R) \subseteq U \)

lemma (in pmetric_space) zero_dist_same_open:

assumes \( r \text{ down-directs } L_+ \), \( U\in \tau \), \( x\in U \), \( y\in X \), \( d\langle x,y\rangle = 0 \)

shows \( y\in U \)

theorem (in pmetric_space) metric_top_carrier:

assumes \( r \text{ down-directs } L_+ \)

shows \( \bigcup \tau = X \)

lemma Top_1_1_L1:

assumes \( T \text{ is }T_0 \) and \( x \in \bigcup T \), \( y \in \bigcup T \) and \( \forall U\in T.\ (x\in U \longleftrightarrow y\in U) \)

shows \( x=y \)

Definition of IsAmetric: \( \text{IsAmetric}(d,X,G,A,r) \equiv \) \( \text{IsApseudoMetric}(d,X,G,A,r) \wedge (\forall x\in X.\ \forall y\in X.\ d\langle x,y\rangle = \text{ TheNeutralElement}(G,A) \longrightarrow x=y) \)

lemma (in metric_space) dist_pos:

assumes \( x\in X \), \( y\in X \), \( x\neq y \)

shows \( 0 \lt d\langle x,y\rangle \), \( d\langle x,y\rangle \in L_+ \)

lemma (in loop1) ls_other_side:

assumes \( x \lt y \)

shows \( 0 \lt - x + y \), \( ( - x + y) \in L_+ \), \( 0 \lt y - x \), \( (y - x) \in L_+ \)

lemma (in pmetric_space) pmetric_loop_valued:

assumes \( x\in X \), \( y\in X \)

shows \( d\langle x,y\rangle \in L^+ \), \( d\langle x,y\rangle \in L \)

lemma (in metric_space) dist_pos:

assumes \( x\in X \), \( y\in X \), \( x\neq y \)

shows \( 0 \lt d\langle x,y\rangle \), \( d\langle x,y\rangle \in L_+ \)

lemma (in loop1) add_subtract_pos:

assumes \( x\in L \), \( 0 \lt y \)

shows \( x - y \lt x \), \( ( - y + x) \lt x \), \( x \lt x + y \), \( x \lt y + x \)

lemma (in pmetric_space) disjoint_disks:

assumes \( x\in X \), \( y\in X \), \( r_x \lt d\langle x,y\rangle \)

shows \( ( - r_x + (d\langle x,y\rangle )) \in L_+ \) and \( disk(x,r_x)\cap disk(y, - r_x + (d\langle x,y\rangle )) = 0 \)

lemma exist2_subset:

assumes \( A\subseteq B \) and \( \exists x\in A.\ \exists y\in A.\ \phi (x,y) \)

shows \( \exists x\in B.\ \exists y\in B.\ \phi (x,y) \)

Definition of isT2: \( T \text{ is }T_2 \equiv \forall x y.\ ((x \in \bigcup T \wedge y \in \bigcup T \wedge x\neq y) \longrightarrow \) \( (\exists U\in T.\ \exists V\in T.\ x\in U \wedge y\in V \wedge U\cap V=0)) \)

Definition of UniformGauge: \( \text{UniformGauge}(X,L,A,r,d) \equiv \{d^{-1}(\{c\in \text{Nonnegative}(L,A,r).\ \langle c,b\rangle \in r\}).\ b\in \text{PositiveSet}(L,A,r)\} \)

lemma (in pmetric_space) uniform_gauge_def_alt: shows \( \mathfrak{B} = \{d^{-1}(\{c\in L^+.\ c\leq b\}).\ b\in L_+\} \)

lemma func1_1_L3:

assumes \( f:X\rightarrow Y \)

shows \( f^{-1}(D) \subseteq X \)

lemma func1_1_L15:

assumes \( f:X\rightarrow Y \)

shows \( f^{-1}(A) = \{x\in X.\ f(x) \in A\} \)

lemma (in pmetric_space) uniform_gauge_relations:

assumes \( B\in \mathfrak{B} \)

shows \( B\subseteq X\times X \)

lemma vimage_mono1:

assumes \( A\subseteq B \)

shows \( f^{-1}(A) \subseteq f^{-1}(B) \)

Definition of IsAnOrdLoop: \( \text{IsAnOrdLoop}(L,A,r) \equiv \) \( \text{IsAloop}(L,A) \wedge r\subseteq L\times L \wedge \text{IsPartOrder}(L,r) \wedge (\forall x\in L.\ \forall y\in L.\ \forall z\in L.\ \) \( ((\langle x,y\rangle \in r \longleftrightarrow \langle A\langle x,z\rangle ,A\langle y,z\rangle \rangle \in r) \wedge (\langle x,y\rangle \in r \longleftrightarrow \langle A\langle z,x\rangle ,A\langle z,y\rangle \rangle \in r ))) \)

Definition of IsPartOrder: \( \text{IsPartOrder}(X,r) \equiv \text{refl}(X,r) \wedge \text{antisym}(r) \wedge \text{trans}(r) \)

lemma (in pmetric_space) uniform_gauge_mono:

assumes \( b_1\leq b_2 \)

shows \( d^{-1}(\{c\in L^+.\ c\leq b_1\}) \subseteq d^{-1}(\{c\in L^+.\ c\leq b_2\}) \)

lemma (in loop1) loop_zero_nonneg: shows \( 0 \in L^+ \)

lemma symm_vimage_symm:

assumes \( f:X\times X\rightarrow Y \) and \( \forall x\in X.\ \forall y\in X.\ f\langle x,y\rangle = f\langle y,x\rangle \)

shows \( f^{-1}(A) = converse(f^{-1}(A)) \)

lemma (in pmetric_space) gauge_symmetric:

assumes \( B\in \mathfrak{B} \)

shows \( B = converse(B) \)

Definition of IsHalfable: \( \text{IsHalfable}(L,A,r) \equiv \forall b_1\in \text{PositiveSet}(L,A,r).\ \exists b_2\in \text{PositiveSet}(L,A,r).\ \langle A\langle b_2,b_2\rangle ,b_1\rangle \in r \)

lemma rel_compdef: shows \( \langle x,z\rangle \in r\circ s \longleftrightarrow (\exists y.\ \langle x,y\rangle \in s \wedge \langle y,z\rangle \in r) \)

lemma (in loop1) add_ineq:

assumes \( x\leq y \), \( z\leq t \)

shows \( x + z \leq y + t \)

lemma (in loop1) loop_ord_trans:

assumes \( x\leq y \) and \( y\leq z \)

shows \( x\leq z \)

lemma (in pmetric_space) gauge_members:

assumes \( x\in X \), \( y\in X \), \( d\langle x,y\rangle \leq b \)

shows \( \langle x,y\rangle \in d^{-1}(\{c\in L^+.\ c\leq b\}) \)

lemma (in pmetric_space) is_halfable_def_alt:

assumes \( \text{IsHalfable}(L,A,r) \), \( b_1\in L_+ \)

shows \( \exists b_2\in L_+.\ b_2 + b_2 \leq b_1 \)

lemma (in pmetric_space) half_vimage_square:

assumes \( b_2\in L_+ \) and \( b_2 + b_2 \leq b_1 \)

defines \( B_1 \equiv d^{-1}(\{c\in L^+.\ c\leq b_1\}) \) and \( B_2 \equiv d^{-1}(\{c\in L^+.\ c\leq b_2\}) \)

shows \( B_2\circ B_2 \subseteq B_1 \)

lemma (in pmetric_space) gauge_1st_cond:

assumes \( r \text{ down-directs } L_+ \), \( B_1\in \mathfrak{B} \), \( B_2\in \mathfrak{B} \)

shows \( \exists B_3\in \mathfrak{B} .\ B_3\subseteq B_1\cap B_2 \)

lemma (in pmetric_space) gauge_2nd_cond:

assumes \( B\in \mathfrak{B} \)

shows \( id(X)\subseteq B \)

corollary (in pmetric_space) gauge_3rd_cond:

assumes \( B_1\in \mathfrak{B} \)

shows \( \exists B_2\in \mathfrak{B} .\ B_2 \subseteq converse(B_1) \)

lemma (in pmetric_space) gauge_4thCond:

assumes \( \text{IsHalfable}(L,A,r) \), \( B_1\in \mathfrak{B} \)

shows \( \exists B_2\in \mathfrak{B} .\ B_2\circ B_2 \subseteq B_1 \)

lemma (in pmetric_space) gauge_5thCond: shows \( \mathfrak{B} \subseteq Pow(X\times X) \)

lemma (in pmetric_space) gauge_6thCond:

assumes \( L_+\neq \emptyset \)

shows \( \mathfrak{B} \neq \emptyset \)

theorem uniformity_base_is_base:

assumes \( X\neq \emptyset \) and \( \mathfrak{B} \text{ is a uniform base on } X \)

shows \( \text{Supersets}(X\times X,\mathfrak{B} ) \text{ is a uniformity on } X \)

theorem uniform_top_is_top:

assumes \( \Phi \text{ is a uniformity on } X \)

shows \( \text{UniformTopology}(\Phi ,X) \text{ is a topology } \) and \( \bigcup \text{UniformTopology}(\Phi ,X) = X \)

Definition of IsUniformityBaseOn: \( \mathfrak{B} \text{ is a uniform base on } X \equiv \) \( (\forall B_1\in \mathfrak{B} .\ \forall B_2\in \mathfrak{B} .\ \exists B_3\in \mathfrak{B} .\ B_3\subseteq B_1\cap B_2) \wedge (\forall B\in \mathfrak{B} .\ id(X)\subseteq B) \wedge \) \( (\forall B_1\in \mathfrak{B} .\ \exists B_2\in \mathfrak{B} .\ B_2 \subseteq converse(B_1)) \wedge (\forall B_1\in \mathfrak{B} .\ \exists B_2\in \mathfrak{B} .\ B_2\circ B_2 \subseteq B_1) \wedge \) \( \mathfrak{B} \subseteq Pow(X\times X) \wedge \mathfrak{B} \neq \emptyset \)

theorem (in rpmetric_space) metric_gauge_base:

assumes \( X\neq \emptyset \)

shows \( \mathfrak{U} \text{ is a uniform base on } X \), \( \text{Supersets}(X\times X,\mathfrak{U} ) \text{ is a uniformity on } X \), \( \text{UniformTopology}( \text{Supersets}(X\times X,\mathfrak{U} ),X) \text{ is a topology } \), \( \bigcup \text{UniformTopology}( \text{Supersets}(X\times X,\mathfrak{U} ),X) = X \)

Definition of UniformTopology: \( \text{UniformTopology}(\Phi ,X) \equiv \{U\in Pow(X).\ \forall x\in U.\ U\in \{V\{x\}.\ V\in \Phi \}\} \)

theorem (in rpmetric_space) metric_gauge_base:

assumes \( X\neq \emptyset \)

Definition of Supersets: \( \text{Supersets}(X,\mathcal{A} ) \equiv \{B\in Pow(X).\ \exists A\in \mathcal{A} .\ A\subseteq B\} \)

lemma (in pmetric_space) disk_in_gauge:

assumes \( b\in L_+ \), \( x\in X \)

defines \( B \equiv d^{-1}(\{c\in L^+.\ c\leq b\}) \)

shows \( B\{x\} = \{y\in X.\ d\langle x,y\rangle \leq b\} \) and \( disk(x,b) \subseteq B\{x\} \)

lemma (in pmetric_space) disks_open:

assumes \( c\in X \), \( R\in L_+ \), \( r \text{ down-directs } L_+ \)

shows \( disk(c,R) \in \tau \)

lemma (in pmetric_space) topology0_valid_in_pmetric_space: