diff --git a/TODO.md b/TODO.md
index dd67a3d6..e84103eb 100644
--- a/TODO.md
+++ b/TODO.md
@@ -1,15 +1,17 @@
 # half-edge
 
-[half-edge] `PortEntry` -- `connection` through `HalfEdge`
+[half-edge] rename `net.activeEdges` to `net.activePortPairs`
 
 [half-edge] `Edge` has two `HalfEdge`s
 
-[half-edge] `HalfEdge` should be a kind of `Value`
+[half-edge] `HalfEdge` should be a kind of `Value` -- `Port` should quit being a `Value`
 
 [half-edge] avoid using `closeAllFreePorts`
 
 [half-edge] `@edge` as a builtin to create two `HalfEdge`s
 
+[half-edge] refactor
+
 # docs
 
 [docs] add mimor to inet docs
diff --git a/src/lang/builtins/connect.ts b/src/lang/builtins/connect.ts
index 409f6009..9ae96d74 100644
--- a/src/lang/builtins/connect.ts
+++ b/src/lang/builtins/connect.ts
@@ -1,5 +1,5 @@
 import { ComposeOptions } from "../compose/compose"
-import { connect } from "../connect/connect"
+import { connectPorts } from "../connect/connectPorts"
 import { Env } from "../env"
 import { unifyTypes } from "../unify/unifyTypes"
 import { formatValue } from "../value/formatValue"
@@ -46,7 +46,7 @@ export function compose(env: Env, options: ComposeOptions): void {
     )
   }
 
-  connect(env.net, first, second)
+  connectPorts(env.net, first, second)
   if (options.checking) {
     unifyTypes(env, options.checking.substitution, first.t, second.t)
   }
diff --git a/src/lang/cap/capInputPort.ts b/src/lang/cap/capInputPort.ts
index 2d467a1f..c2ed9f8a 100644
--- a/src/lang/cap/capInputPort.ts
+++ b/src/lang/cap/capInputPort.ts
@@ -1,4 +1,4 @@
-import { connect } from "../connect/connect"
+import { connectPorts } from "../connect/connectPorts"
 import { Mod } from "../mod"
 import { Net } from "../net"
 import { addNode } from "../net/addNode"
@@ -23,6 +23,6 @@ export function capInputPort(mod: Mod, net: Net, port: Port): Port {
   }
 
   const capPort = findOutputPorts(net, node)[0]
-  connect(net, port, capPort)
+  connectPorts(net, port, capPort)
   return capPort
 }
diff --git a/src/lang/cap/capOutputPort.ts b/src/lang/cap/capOutputPort.ts
index e43c9be8..20f03ba4 100644
--- a/src/lang/cap/capOutputPort.ts
+++ b/src/lang/cap/capOutputPort.ts
@@ -1,4 +1,4 @@
-import { connect } from "../connect/connect"
+import { connectPorts } from "../connect/connectPorts"
 import { Mod } from "../mod"
 import { Net } from "../net"
 import { addNode } from "../net/addNode"
@@ -23,6 +23,6 @@ export function capOutputPort(mod: Mod, net: Net, port: Port): Port {
   }
 
   const capPort = findInputPorts(net, node)[0]
-  connect(net, port, capPort)
+  connectPorts(net, port, capPort)
   return capPort
 }
diff --git a/src/lang/check/checkPortSigns.ts b/src/lang/check/checkPortSigns.ts
index 606913de..fb8ac8bd 100644
--- a/src/lang/check/checkPortSigns.ts
+++ b/src/lang/check/checkPortSigns.ts
@@ -6,7 +6,7 @@ export function checkPortSigns(net: Net, first: Port, second: Port): void {
   if (first.sign === 1 && second.sign === 1) {
     throw new Error(
       [
-        `[checkSigns] I expect the two ports to have opposite signs,`,
+        `[checkPortSigns] I expect the two ports to have opposite signs,`,
         `  but they all have positive sign.`,
         ``,
         `  first port: ${formatPort(net, first)}`,
@@ -18,7 +18,7 @@ export function checkPortSigns(net: Net, first: Port, second: Port): void {
   if (first.sign === -1 && second.sign === -1) {
     throw new Error(
       [
-        `[checkSigns] I expect the two ports to have opposite signs,`,
+        `[checkPortSigns] I expect the two ports to have opposite signs,`,
         `  but they all have negative sign.`,
         ``,
         `  first port: ${formatPort(net, first)}`,
diff --git a/src/lang/compose/compose.ts b/src/lang/compose/compose.ts
index 0d239323..a69a44ac 100644
--- a/src/lang/compose/compose.ts
+++ b/src/lang/compose/compose.ts
@@ -1,5 +1,5 @@
 import { Checking } from "../checking"
-import { connect } from "../connect/connect"
+import { connectPorts } from "../connect/connectPorts"
 import { Env } from "../env"
 import { appendReport } from "../errors/appendReport"
 import { Mod } from "../mod"
@@ -142,7 +142,7 @@ export function compose(
           )
         }
 
-        connect(env.net, value, currentPort)
+        connectPorts(env.net, value, currentPort)
         if (options.checking) {
           unifyTypes(env, options.checking.substitution, value.t, currentPort.t)
         }
diff --git a/src/lang/compose/composeNode.ts b/src/lang/compose/composeNode.ts
index 1ea108bd..e843fd61 100644
--- a/src/lang/compose/composeNode.ts
+++ b/src/lang/compose/composeNode.ts
@@ -1,4 +1,4 @@
-import { connect } from "../connect/connect"
+import { connectPorts } from "../connect/connectPorts"
 import { Env } from "../env"
 import { refreshNode } from "../freshen/refreshNode"
 import { findInputPorts } from "../net/findInputPorts"
@@ -39,7 +39,8 @@ export function composeNode(
       )
     }
 
-    connect(env.net, value, port)
+    connectPorts(env.net, value, port)
+
     if (options.checking) {
       unifyTypes(env, options.checking.substitution, value.t, port.t)
     }
diff --git a/src/lang/compose/findCurrentPortOrFail.ts b/src/lang/compose/findCurrentPortOrFail.ts
index 464b83c8..4dca72c6 100644
--- a/src/lang/compose/findCurrentPortOrFail.ts
+++ b/src/lang/compose/findCurrentPortOrFail.ts
@@ -1,4 +1,5 @@
 import { Net } from "../net"
+import { findHalfEdgeEntry } from "../net/findHalfEdgeEntry"
 import { findInputPorts } from "../net/findInputPorts"
 import { findOutputPorts } from "../net/findOutputPorts"
 import { findPortEntry } from "../net/findPortEntry"
@@ -66,15 +67,31 @@ function findPortInNode(
 ): Port | undefined {
   for (const port of findInputPorts(net, node)) {
     if (port.name === portName) {
-      const portEntry = findPortEntry(net, port)
-      return portEntry?.connection?.port
+      return findCorrespondingPort(net, port)
     }
   }
 
   for (const port of findOutputPorts(net, node)) {
     if (port.name === portName) {
-      const portEntry = findPortEntry(net, port)
-      return portEntry?.connection?.port
+      return findCorrespondingPort(net, port)
     }
   }
 }
+
+function findCorrespondingPort(net: Net, port: Port): Port | undefined {
+  const portEntry = findPortEntry(net, port)
+
+  if (portEntry === undefined) return
+  if (portEntry.connection === undefined) return
+
+  const halfEdge = portEntry.connection.halfEdge
+  const halfEdgeEntry = findHalfEdgeEntry(net, halfEdge)
+
+  if (halfEdgeEntry === undefined) return
+
+  const ohterHalfEdgeEntry = findHalfEdgeEntry(net, halfEdgeEntry.otherHalfEdge)
+
+  if (ohterHalfEdgeEntry === undefined) return
+
+  return ohterHalfEdgeEntry.port
+}
diff --git a/src/lang/connect/connectHalfEdges.ts b/src/lang/connect/connectHalfEdges.ts
new file mode 100644
index 00000000..7db08c5d
--- /dev/null
+++ b/src/lang/connect/connectHalfEdges.ts
@@ -0,0 +1,31 @@
+import { HalfEdge } from "../half-edge"
+import { Net } from "../net"
+import { deleteHalfEdgeEntry } from "../net/deleteHalfEdgeEntry"
+import { findHalfEdgeEntryOrFail } from "../net/findHalfEdgeEntryOrFail"
+
+export function connectHalfEdges(
+  net: Net,
+  firstHalfEdge: HalfEdge,
+  secondHalfEdge: HalfEdge,
+): void {
+  const firstHalfEdgeEntry = findHalfEdgeEntryOrFail(net, firstHalfEdge)
+  const secondHalfEdgeEntry = findHalfEdgeEntryOrFail(net, secondHalfEdge)
+
+  const firstOtherHalfEdge = firstHalfEdgeEntry.otherHalfEdge
+  const secondOtherHalfEdge = secondHalfEdgeEntry.otherHalfEdge
+
+  const firstOtherHalfEdgeEntry = findHalfEdgeEntryOrFail(
+    net,
+    firstOtherHalfEdge,
+  )
+  const secondOtherHalfEdgeEntry = findHalfEdgeEntryOrFail(
+    net,
+    secondOtherHalfEdge,
+  )
+
+  firstOtherHalfEdgeEntry.otherHalfEdge = secondOtherHalfEdge
+  secondOtherHalfEdgeEntry.otherHalfEdge = firstOtherHalfEdge
+
+  deleteHalfEdgeEntry(net, firstHalfEdge)
+  deleteHalfEdgeEntry(net, secondHalfEdge)
+}
diff --git a/src/lang/connect/connectPortWithHalfEdge.ts b/src/lang/connect/connectPortWithHalfEdge.ts
new file mode 100644
index 00000000..8e679573
--- /dev/null
+++ b/src/lang/connect/connectPortWithHalfEdge.ts
@@ -0,0 +1,42 @@
+import { edgeEqual } from "../edge"
+import { HalfEdge } from "../half-edge"
+import { Net } from "../net"
+import { findHalfEdgeEntryOrFail } from "../net/findHalfEdgeEntryOrFail"
+import { findPortRecordOrFail } from "../net/findPortRecordOrFail"
+import { Port } from "../port"
+
+export function connectPortWithHalfEdge(
+  net: Net,
+  port: Port,
+  halfEdge: HalfEdge,
+): void {
+  const portRecord = findPortRecordOrFail(net, port.node)
+  portRecord[port.name].connection = { halfEdge }
+
+  const halfEdgeEntry = findHalfEdgeEntryOrFail(net, halfEdge)
+  if (halfEdgeEntry.port !== undefined) {
+    throw new Error(
+      `[connectPortWithHalfEdge] halfEdgeEntry is already connected`,
+    )
+  }
+
+  halfEdgeEntry.port = port
+
+  const otherHalfEdgeEntry = findHalfEdgeEntryOrFail(
+    net,
+    halfEdgeEntry.otherHalfEdge,
+  )
+
+  if (otherHalfEdgeEntry.port !== undefined) {
+    if (port.isPrincipal && otherHalfEdgeEntry.port.isPrincipal) {
+      const edge = {
+        first: port,
+        second: otherHalfEdgeEntry.port,
+      }
+
+      if (!net.activeEdges.find((activeEdge) => edgeEqual(activeEdge, edge))) {
+        net.activeEdges.push(edge)
+      }
+    }
+  }
+}
diff --git a/src/lang/connect/connect.ts b/src/lang/connect/connectPorts.ts
similarity index 55%
rename from src/lang/connect/connect.ts
rename to src/lang/connect/connectPorts.ts
index f2d72851..96ad546f 100644
--- a/src/lang/connect/connect.ts
+++ b/src/lang/connect/connectPorts.ts
@@ -1,11 +1,12 @@
 import { checkPortSigns } from "../check/checkPortSigns"
 import { Net } from "../net"
+import { addEdge } from "../net/addEdge"
 import { findPortEntry } from "../net/findPortEntry"
-import { findPortRecordOrFail } from "../net/findPortRecordOrFail"
 import { Port } from "../port"
 import { formatPort } from "../port/formatPort"
+import { connectPortWithHalfEdge } from "./connectPortWithHalfEdge"
 
-export function connect(net: Net, first: Port, second: Port): void {
+export function connectPorts(net: Net, first: Port, second: Port): void {
   const firstPortEntry = findPortEntry(net, first)
 
   if (firstPortEntry?.connection !== undefined) {
@@ -14,10 +15,6 @@ export function connect(net: Net, first: Port, second: Port): void {
         `[connect] The first port is already connected.`,
         ``,
         `  first port: ${formatPort(net, first)}`,
-        `  first connected port: ${formatPort(
-          net,
-          firstPortEntry.connection.port,
-        )}`,
         `  second port: ${formatPort(net, second)}`,
       ].join("\n"),
     )
@@ -32,25 +29,14 @@ export function connect(net: Net, first: Port, second: Port): void {
         ``,
         `  first port: ${formatPort(net, first)}`,
         `  second port: ${formatPort(net, second)}`,
-        `  second connected port: ${formatPort(
-          net,
-          secondPortEntry.connection.port,
-        )}`,
       ].join("\n"),
     )
   }
 
   checkPortSigns(net, first, second)
 
-  const edge = { first, second }
+  const halfEdges = addEdge(net)
 
-  const firstPortRecord = findPortRecordOrFail(net, first.node)
-  firstPortRecord[first.name].connection = { port: second }
-
-  const secondPortRecord = findPortRecordOrFail(net, second.node)
-  secondPortRecord[second.name].connection = { port: first }
-
-  if (first.isPrincipal && second.isPrincipal) {
-    net.activeEdges.push(edge)
-  }
+  connectPortWithHalfEdge(net, first, halfEdges.first)
+  connectPortWithHalfEdge(net, second, halfEdges.second)
 }
diff --git a/src/lang/half-edge/createHalfEdgeId.ts b/src/lang/half-edge/createHalfEdgeId.ts
index 5862c587..a9a1d58c 100644
--- a/src/lang/half-edge/createHalfEdgeId.ts
+++ b/src/lang/half-edge/createHalfEdgeId.ts
@@ -1,6 +1,6 @@
-import { Mod } from "../mod"
+import { globalHalfEdgeInfo } from "./globalHalfEdgeInfo"
 
-export function createHalfEdgeId(mod: Mod): string {
-  const n = mod.halfEdgeCounter++
+export function createHalfEdgeId(): string {
+  const n = globalHalfEdgeInfo.counter++
   return n.toString()
 }
diff --git a/src/lang/half-edge/globalHalfEdgeInfo.ts b/src/lang/half-edge/globalHalfEdgeInfo.ts
new file mode 100644
index 00000000..09d08d88
--- /dev/null
+++ b/src/lang/half-edge/globalHalfEdgeInfo.ts
@@ -0,0 +1,3 @@
+export const globalHalfEdgeInfo = {
+  counter: 0,
+}
diff --git a/src/lang/interact/interact.ts b/src/lang/interact/interact.ts
index d5adf533..5ee4f523 100644
--- a/src/lang/interact/interact.ts
+++ b/src/lang/interact/interact.ts
@@ -3,7 +3,9 @@ import { compose } from "../compose/compose"
 import { Edge } from "../edge"
 import { Env } from "../env"
 import { findRuleByPorts } from "../mod/findRuleByPorts"
+import { deleteHalfEdgeEntry } from "../net/deleteHalfEdgeEntry"
 import { deleteNodeEntry } from "../net/deleteNodeEntry"
+import { findPortEntryOrFail } from "../net/findPortEntryOrFail"
 
 export type InteractOptions = {
   checking?: Checking
@@ -15,6 +17,7 @@ export function interact(
   options: InteractOptions,
 ): void {
   const rule = findRuleByPorts(env.mod, activeEdge.first, activeEdge.second)
+
   if (rule === undefined) return
 
   for (const word of rule.words) {
@@ -27,6 +30,17 @@ export function interact(
     })
   }
 
+  const firstPortEntry = findPortEntryOrFail(env.net, activeEdge.first)
+  const secondPortEntry = findPortEntryOrFail(env.net, activeEdge.second)
+
+  if (firstPortEntry.connection !== undefined) {
+    deleteHalfEdgeEntry(env.net, firstPortEntry.connection.halfEdge)
+  }
+
+  if (secondPortEntry.connection !== undefined) {
+    deleteHalfEdgeEntry(env.net, secondPortEntry.connection.halfEdge)
+  }
+
   deleteNodeEntry(env.net, activeEdge.first.node)
   deleteNodeEntry(env.net, activeEdge.second.node)
 }
diff --git a/src/lang/mod/Mod.ts b/src/lang/mod/Mod.ts
index 2fa60ac0..09ae0855 100644
--- a/src/lang/mod/Mod.ts
+++ b/src/lang/mod/Mod.ts
@@ -16,6 +16,4 @@ export type Mod = {
   builtins: Map<string, Definition>
   ruleEntries: Map<string, RuleEntry>
   requiredMods: Map<string, Mod>
-  nodeCounters: Map<string, number>
-  halfEdgeCounter: number
 }
diff --git a/src/lang/mod/createMod.ts b/src/lang/mod/createMod.ts
index 2c026a46..f4a58d34 100644
--- a/src/lang/mod/createMod.ts
+++ b/src/lang/mod/createMod.ts
@@ -20,8 +20,6 @@ export function createMod(options: {
     builtins: new Map(),
     ruleEntries: new Map(),
     requiredMods: new Map(),
-    nodeCounters: new Map(),
-    halfEdgeCounter: 0,
   } as Mod
 
   mod.env = createEnv(mod)
diff --git a/src/lang/mod/findRuleByNodes.ts b/src/lang/mod/findRuleByNodes.ts
index 6690e388..fa612a98 100644
--- a/src/lang/mod/findRuleByNodes.ts
+++ b/src/lang/mod/findRuleByNodes.ts
@@ -11,7 +11,8 @@ export function findRuleByNodes(
   const firstKey = nodeKeyWithoutId(firstNode)
   const secondKey = nodeKeyWithoutId(secondNode)
 
-  const key = `${firstKey} ${secondKey}`
-
-  return mod.ruleEntries.get(key)
+  return (
+    mod.ruleEntries.get(`${firstKey} ${secondKey}`) ||
+    mod.ruleEntries.get(`${secondKey} ${firstKey}`)
+  )
 }
diff --git a/src/lang/net/Net.ts b/src/lang/net/Net.ts
index 42a32343..14863b8c 100644
--- a/src/lang/net/Net.ts
+++ b/src/lang/net/Net.ts
@@ -23,7 +23,7 @@ export type Net = {
 export type HalfEdgeEntry = {
   id: string
   port?: Port
-  otherHalf?: HalfEdge
+  otherHalfEdge: HalfEdge
 }
 
 export type NodeEntry = {
@@ -56,6 +56,5 @@ export type PortEntry = {
 //   interfere with each other.
 
 export type Connection = {
-  // halfEdge: HalfEdge
-  port: Port
+  halfEdge: HalfEdge
 }
diff --git a/src/lang/net/addEdge.ts b/src/lang/net/addEdge.ts
index 7e789056..0f37f3eb 100644
--- a/src/lang/net/addEdge.ts
+++ b/src/lang/net/addEdge.ts
@@ -1,17 +1,16 @@
 import { HalfEdge } from "../half-edge"
 import { createHalfEdgeId } from "../half-edge/createHalfEdgeId"
-import { Mod } from "../mod"
-import { Net } from "../net"
+import { HalfEdgeEntry, Net } from "../net"
 
-export function addEdge(
-  net: Net,
-  mod: Mod,
-): { first: HalfEdge; second: HalfEdge } {
-  const firstId = createHalfEdgeId(mod)
-  const first = { id: firstId }
+export function addEdge(net: Net): { first: HalfEdge; second: HalfEdge } {
+  const firstId = createHalfEdgeId()
+  const first = { id: firstId } as HalfEdgeEntry
 
-  const secondId = createHalfEdgeId(mod)
-  const second = { id: secondId }
+  const secondId = createHalfEdgeId()
+  const second = { id: secondId } as HalfEdgeEntry
+
+  first.otherHalfEdge = { id: second.id }
+  second.otherHalfEdge = { id: first.id }
 
   net.halfEdgeEntries.set(firstId, first)
   net.halfEdgeEntries.set(secondId, second)
diff --git a/src/lang/net/addNode.ts b/src/lang/net/addNode.ts
index fc2e21f6..3c2c65bb 100644
--- a/src/lang/net/addNode.ts
+++ b/src/lang/net/addNode.ts
@@ -12,7 +12,7 @@ export function addNode(
   input: Array<PortExp>,
   output: Array<PortExp>,
 ): Node {
-  const id = createNodeId(mod, name)
+  const id = createNodeId(name)
 
   const node: Node = {
     "@type": "Value",
diff --git a/src/lang/net/allEdges.ts b/src/lang/net/allEdges.ts
index 8bdacf33..102c905a 100644
--- a/src/lang/net/allEdges.ts
+++ b/src/lang/net/allEdges.ts
@@ -1,9 +1,9 @@
 import { Edge } from "../edge"
 import { nodeKey } from "../node/nodeKey"
-import { Port } from "../port"
 import { Net } from "./Net"
 import { createNodeFromNodeEntry } from "./createNodeFromNodeEntry"
 import { createPortFromPortEntry } from "./createPortFromPortEntry"
+import { findHalfEdgeEntryOrFail } from "./findHalfEdgeEntryOrFail"
 
 export function allEdges(net: Net): Array<Edge> {
   const edges: Array<Edge> = []
@@ -14,8 +14,22 @@ export function allEdges(net: Net): Array<Edge> {
 
     for (const portEntry of Object.values(nodeEntry.ports)) {
       if (portEntry.connection) {
-        const second = portEntry.connection.port
-        const first: Port = createPortFromPortEntry(node, portEntry)
+        const first = createPortFromPortEntry(node, portEntry)
+
+        const halfEdgeEntry = findHalfEdgeEntryOrFail(
+          net,
+          portEntry.connection.halfEdge,
+        )
+
+        const otherHalfEdgeEntry = findHalfEdgeEntryOrFail(
+          net,
+          halfEdgeEntry.otherHalfEdge,
+        )
+
+        const second = otherHalfEdgeEntry.port
+        if (second === undefined) {
+          continue
+        }
 
         const firstOccur = `${nodeKey(node)}-${portEntry.name}`
         const secondOccur = `${nodeKey(second.node)}-${second.name}`
diff --git a/src/lang/net/copyConnectedComponent.ts b/src/lang/net/copyConnectedComponent.ts
index 7a34c6a6..86fe023d 100644
--- a/src/lang/net/copyConnectedComponent.ts
+++ b/src/lang/net/copyConnectedComponent.ts
@@ -1,7 +1,9 @@
 import { edgeEqual } from "../edge/edgeEqual"
-import { Node } from "../node"
+import { Node, nodeKey } from "../node"
 import { Net } from "./Net"
-import { copyNode } from "./copyNode"
+import { cloneNodeEntry } from "./cloneNodeEntry"
+import { findHalfEdgeEntryOrFail } from "./findHalfEdgeEntryOrFail"
+import { findNodeEntryOrFail } from "./findNodeEntryOrFail"
 import { findPortRecordOrFail } from "./findPortRecordOrFail"
 import { hasNode } from "./hasNode"
 
@@ -16,11 +18,32 @@ export function copyConnectedComponent(
 
   const portRecord = findPortRecordOrFail(net, node)
 
-  copyNode(net, component, node)
+  const entry = findNodeEntryOrFail(net, node)
+  component.nodeEntries.set(nodeKey(node), cloneNodeEntry(entry))
 
   for (const portEntry of Object.values(portRecord)) {
     if (portEntry.connection) {
-      copyConnectedComponent(net, component, portEntry.connection.port.node)
+      const halfEdgeEntry = findHalfEdgeEntryOrFail(
+        net,
+        portEntry.connection.halfEdge,
+      )
+
+      const otherHalfEdgeEntry = findHalfEdgeEntryOrFail(
+        net,
+        halfEdgeEntry.otherHalfEdge,
+      )
+
+      component.halfEdgeEntries.set(halfEdgeEntry.id, {
+        ...halfEdgeEntry,
+      })
+
+      component.halfEdgeEntries.set(otherHalfEdgeEntry.id, {
+        ...otherHalfEdgeEntry,
+      })
+
+      if (otherHalfEdgeEntry.port) {
+        copyConnectedComponent(net, component, otherHalfEdgeEntry.port.node)
+      }
     }
   }
 
diff --git a/src/lang/net/copyNode.ts b/src/lang/net/copyNode.ts
deleted file mode 100644
index 47c108a9..00000000
--- a/src/lang/net/copyNode.ts
+++ /dev/null
@@ -1,10 +0,0 @@
-import { Node } from "../node"
-import { nodeKey } from "../node/nodeKey"
-import { Net } from "./Net"
-import { cloneNodeEntry } from "./cloneNodeEntry"
-import { findNodeEntryOrFail } from "./findNodeEntryOrFail"
-
-export function copyNode(source: Net, target: Net, node: Node): void {
-  const entry = findNodeEntryOrFail(source, node)
-  target.nodeEntries.set(nodeKey(node), cloneNodeEntry(entry))
-}
diff --git a/src/lang/net/deleteHalfEdgeEntry.ts b/src/lang/net/deleteHalfEdgeEntry.ts
new file mode 100644
index 00000000..3a5320c2
--- /dev/null
+++ b/src/lang/net/deleteHalfEdgeEntry.ts
@@ -0,0 +1,6 @@
+import { HalfEdge } from "../half-edge"
+import { Net } from "./Net"
+
+export function deleteHalfEdgeEntry(net: Net, halfEdge: HalfEdge): void {
+  net.halfEdgeEntries.delete(halfEdge.id)
+}
diff --git a/src/lang/net/disconnectPort.ts b/src/lang/net/disconnectPort.ts
index 96c4b7ae..379242a7 100644
--- a/src/lang/net/disconnectPort.ts
+++ b/src/lang/net/disconnectPort.ts
@@ -1,19 +1,25 @@
 import { Port } from "../port"
 import { Net } from "./Net"
+import { findHalfEdgeEntry } from "./findHalfEdgeEntry"
 import { findNodeEntry } from "./findNodeEntry"
 
-export function disconnectPort(net: Net, port: Port): Port | undefined {
+export function disconnectPort(net: Net, port: Port): void {
   const nodeEntry = findNodeEntry(net, port.node)
   if (nodeEntry === undefined) {
     return undefined
   }
 
-  const connectedPort = nodeEntry.ports[port.name].connection?.port
-  delete nodeEntry.ports[port.name].connection
+  const portEntry = nodeEntry.ports[port.name]
 
-  if (connectedPort) {
-    disconnectPort(net, connectedPort)
-  }
+  const halfEdge = portEntry.connection?.halfEdge
+  delete portEntry.connection
+
+  if (halfEdge !== undefined) {
+    const halfEdgeEntry = findHalfEdgeEntry(net, halfEdge)
+    if (halfEdgeEntry === undefined) {
+      throw new Error(`[disconnectPort] Fail to find halfEdgeEntry`)
+    }
 
-  return connectedPort
+    delete halfEdgeEntry.port
+  }
 }
diff --git a/src/lang/net/findHalfEdgeEntry.ts b/src/lang/net/findHalfEdgeEntry.ts
new file mode 100644
index 00000000..2913e63a
--- /dev/null
+++ b/src/lang/net/findHalfEdgeEntry.ts
@@ -0,0 +1,9 @@
+import { HalfEdge } from "../half-edge"
+import { HalfEdgeEntry, Net } from "./Net"
+
+export function findHalfEdgeEntry(
+  net: Net,
+  halfEdge: HalfEdge,
+): HalfEdgeEntry | undefined {
+  return net.halfEdgeEntries.get(halfEdge.id)
+}
diff --git a/src/lang/net/findHalfEdgeEntryOrFail.ts b/src/lang/net/findHalfEdgeEntryOrFail.ts
new file mode 100644
index 00000000..894c4b38
--- /dev/null
+++ b/src/lang/net/findHalfEdgeEntryOrFail.ts
@@ -0,0 +1,15 @@
+import { HalfEdge } from "../half-edge"
+import { HalfEdgeEntry, Net } from "./Net"
+import { findHalfEdgeEntry } from "./findHalfEdgeEntry"
+
+export function findHalfEdgeEntryOrFail(
+  net: Net,
+  halfEdge: HalfEdge,
+): HalfEdgeEntry {
+  const halfEdgeEntry = findHalfEdgeEntry(net, halfEdge)
+  if (halfEdgeEntry === undefined) {
+    throw new Error(`[findHalfEdgeEntryOrFail] Undefined halfEdgeEntry`)
+  }
+
+  return halfEdgeEntry
+}
diff --git a/src/lang/net/findPortEntryOrFail.ts b/src/lang/net/findPortEntryOrFail.ts
new file mode 100644
index 00000000..934d2c14
--- /dev/null
+++ b/src/lang/net/findPortEntryOrFail.ts
@@ -0,0 +1,12 @@
+import { Port } from "../port"
+import { Net, PortEntry } from "./Net"
+import { findPortEntry } from "./findPortEntry"
+
+export function findPortEntryOrFail(net: Net, port: Port): PortEntry {
+  const portEntry = findPortEntry(net, port)
+  if (portEntry === undefined) {
+    throw new Error(`[findPortEntryOrFail] Undefined port`)
+  }
+
+  return portEntry
+}
diff --git a/src/lang/net/findPortRecordOrFail.ts b/src/lang/net/findPortRecordOrFail.ts
index 6d68db7b..cb001aa7 100644
--- a/src/lang/net/findPortRecordOrFail.ts
+++ b/src/lang/net/findPortRecordOrFail.ts
@@ -8,7 +8,7 @@ export function findPortRecordOrFail(net: Net, node: Node): PortRecord {
   if (nodeEntry === undefined) {
     throw new Error(
       [
-        `[findNodePortsOrFail] I can not find node entry for node.`,
+        `[findPortRecordOrFail] I can not find node entry for node.`,
         ``,
         `  node: ${formatNode(net, node)}`,
       ].join("\n"),
diff --git a/src/lang/net/moveConnectedComponent.ts b/src/lang/net/moveConnectedComponent.ts
index 9c1943cc..4bc2b90f 100644
--- a/src/lang/net/moveConnectedComponent.ts
+++ b/src/lang/net/moveConnectedComponent.ts
@@ -1,30 +1,61 @@
-import { Edge } from "../edge"
-import { edgeEqual } from "../edge/edgeEqual"
-import { Node } from "../node"
+import { Edge, edgeEqual } from "../edge"
+import { Node, nodeKey } from "../node"
 import { Net } from "./Net"
+import { findHalfEdgeEntryOrFail } from "./findHalfEdgeEntryOrFail"
+import { findNodeEntryOrFail } from "./findNodeEntryOrFail"
 import { findPortRecordOrFail } from "./findPortRecordOrFail"
 import { hasNode } from "./hasNode"
-import { moveNode } from "./moveNode"
 
 export function moveConnectedComponent(
   net: Net,
   component: Net,
   node: Node,
 ): void {
+  moveNodeEntries(net, component, node)
+  deleteHalfEdgeEntries(net, component)
+  moveActiveEdges(net, component)
+}
+
+export function deleteHalfEdgeEntries(net: Net, component: Net): void {
+  for (const halfEdgeEntry of component.halfEdgeEntries.values()) {
+    net.halfEdgeEntries.delete(halfEdgeEntry.id)
+  }
+}
+
+export function moveNodeEntries(net: Net, component: Net, node: Node): void {
   if (hasNode(component, node)) {
     return
   }
 
   const portRecord = findPortRecordOrFail(net, node)
 
-  moveNode(net, component, node)
+  const entry = findNodeEntryOrFail(net, node)
+  component.nodeEntries.set(nodeKey(node), entry)
+  net.nodeEntries.delete(nodeKey(node))
 
   for (const portEntry of Object.values(portRecord)) {
     if (portEntry.connection) {
-      moveConnectedComponent(net, component, portEntry.connection.port.node)
+      const halfEdgeEntry = findHalfEdgeEntryOrFail(
+        net,
+        portEntry.connection.halfEdge,
+      )
+
+      const ohterHalfEdgeEntry = findHalfEdgeEntryOrFail(
+        net,
+        halfEdgeEntry.otherHalfEdge,
+      )
+
+      component.halfEdgeEntries.set(halfEdgeEntry.id, halfEdgeEntry)
+      component.halfEdgeEntries.set(ohterHalfEdgeEntry.id, ohterHalfEdgeEntry)
+
+      if (ohterHalfEdgeEntry.port) {
+        moveNodeEntries(net, component, ohterHalfEdgeEntry.port.node)
+      }
     }
   }
+}
 
+export function moveActiveEdges(net: Net, component: Net): void {
   const remainingActiveEdges: Array<Edge> = []
 
   for (const activeEdge of net.activeEdges) {
diff --git a/src/lang/net/moveNode.ts b/src/lang/net/moveNode.ts
deleted file mode 100644
index 4f66a096..00000000
--- a/src/lang/net/moveNode.ts
+++ /dev/null
@@ -1,10 +0,0 @@
-import { Node } from "../node"
-import { nodeKey } from "../node/nodeKey"
-import { Net } from "./Net"
-import { findNodeEntryOrFail } from "./findNodeEntryOrFail"
-
-export function moveNode(source: Net, target: Net, node: Node): void {
-  const entry = findNodeEntryOrFail(source, node)
-  target.nodeEntries.set(nodeKey(node), entry)
-  source.nodeEntries.delete(nodeKey(node))
-}
diff --git a/src/lang/node/createNodeId.ts b/src/lang/node/createNodeId.ts
index beb9971c..c1141a3e 100644
--- a/src/lang/node/createNodeId.ts
+++ b/src/lang/node/createNodeId.ts
@@ -1,12 +1,12 @@
-import { Mod } from "../mod"
+import { globalNodeCounters } from "./globalNodeCounters"
 
-export function createNodeId(mod: Mod, name: string): string {
-  const foundCounter = mod.nodeCounters.get(name)
+export function createNodeId(name: string): string {
+  const foundCounter = globalNodeCounters.get(name)
   if (foundCounter === undefined) {
-    mod.nodeCounters.set(name, 0)
+    globalNodeCounters.set(name, 0)
     return String(0)
   } else {
-    mod.nodeCounters.set(name, foundCounter + 1)
+    globalNodeCounters.set(name, foundCounter + 1)
     return String(foundCounter + 1)
   }
 }
diff --git a/src/lang/node/globalNodeCounters.ts b/src/lang/node/globalNodeCounters.ts
new file mode 100644
index 00000000..5d02fd3a
--- /dev/null
+++ b/src/lang/node/globalNodeCounters.ts
@@ -0,0 +1 @@
+export const globalNodeCounters: Map<string, number> = new Map()
diff --git a/src/lang/present/presentRuleAsNets.ts b/src/lang/present/presentRuleAsNets.ts
index 39b190e7..a7301f6c 100644
--- a/src/lang/present/presentRuleAsNets.ts
+++ b/src/lang/present/presentRuleAsNets.ts
@@ -1,6 +1,6 @@
 import { capNodeNonPrinciplePorts } from "../cap"
 import { compose } from "../compose/compose"
-import { connect } from "../connect/connect"
+import { connectPorts } from "../connect/connectPorts"
 import { createEnv } from "../env/createEnv"
 import { Mod, findDefinitionOrFail } from "../mod"
 import { findRuleByName } from "../mod/findRuleByName"
@@ -64,7 +64,7 @@ function collectInitialNet(net: Net, first: Node, second: Node): Net {
   const firstPrincipalPort = findPrincipalPort(initial, first)
   const secondPrincipalPort = findPrincipalPort(initial, second)
 
-  connect(initial, firstPrincipalPort, secondPrincipalPort)
+  connectPorts(initial, firstPrincipalPort, secondPrincipalPort)
 
   return initial
 }
diff --git a/src/lang/run/collectConnectedPort.ts b/src/lang/run/collectConnectedPort.ts
index cc17b9f0..17c48cbf 100644
--- a/src/lang/run/collectConnectedPort.ts
+++ b/src/lang/run/collectConnectedPort.ts
@@ -1,6 +1,7 @@
 import { Net } from "../net"
 import { deleteNodeEntry } from "../net/deleteNodeEntry"
 import { disconnectPort } from "../net/disconnectPort"
+import { findHalfEdgeEntryOrFail } from "../net/findHalfEdgeEntryOrFail"
 import { findPortEntry } from "../net/findPortEntry"
 import { Port } from "../port"
 import { formatPort } from "../port/formatPort"
@@ -10,14 +11,30 @@ export function collectConnectedPort(component: Net, capPort: Port): Port {
   if (capPortEntry?.connection === undefined) {
     throw new Error(
       [
-        `[run] I expect the capPort to be connected.`,
+        `[collectConnectedPort] I expect the capPort to be connected.`,
         ``,
         `  capPort: ${formatPort(component, capPort)}`,
       ].join("\n"),
     )
   }
 
-  const connectedPort = capPortEntry.connection.port
+  const halfEdgeEntry = findHalfEdgeEntryOrFail(
+    component,
+    capPortEntry.connection.halfEdge,
+  )
+
+  const ohterHalfEdgeEntry = findHalfEdgeEntryOrFail(
+    component,
+    halfEdgeEntry.otherHalfEdge,
+  )
+
+  const connectedPort = ohterHalfEdgeEntry.port
+
+  if (connectedPort === undefined) {
+    throw new Error(
+      `[collectConnectedPort] I expect halfEdgeEntry to have port.`,
+    )
+  }
 
   disconnectPort(component, capPort)
   deleteNodeEntry(component, capPort.node)
diff --git a/tests/checking/begin-sign.error.i.err b/tests/checking/begin-sign.error.i.err
index 3df596ea..68c41627 100644
--- a/tests/checking/begin-sign.error.i.err
+++ b/tests/checking/begin-sign.error.i.err
@@ -1,4 +1,4 @@
-[checkSigns] I expect the two ports to have opposite signs,
+[checkPortSigns] I expect the two ports to have opposite signs,
   but they all have positive sign.
 
   first port: (sole₁)-value!
diff --git a/tests/checking/rule-sign.error.i.err b/tests/checking/rule-sign.error.i.err
index b1b33503..d17eb786 100644
--- a/tests/checking/rule-sign.error.i.err
+++ b/tests/checking/rule-sign.error.i.err
@@ -1,4 +1,4 @@
-[checkSigns] I expect the two ports to have opposite signs,
+[checkPortSigns] I expect the two ports to have opposite signs,
   but they all have positive sign.
 
   first port: (@inputPortCap₂)-covering
diff --git a/tests/datatype/Nat.test.i.out b/tests/datatype/Nat.test.i.out
index 9049e308..3ef82d36 100644
--- a/tests/datatype/Nat.test.i.out
+++ b/tests/datatype/Nat.test.i.out
@@ -25,10 +25,10 @@ netFromPort (add₁₅)-return
   (add1₂₅)-prev value-(zero₂₄)
   (add1₂₄)-prev value-(zero₂₃)
 end
-netFromPort (add1₃₀)-value!
+netFromPort (add1₂₉)-value!
+  (add1₂₉)-prev value-(add1₃₀)
   (add1₃₀)-prev value-(add1₃₁)
-  (add1₃₁)-prev value-(add1₂₈)
-  (add1₂₈)-prev value-(add1₂₄)
+  (add1₃₁)-prev value-(add1₂₄)
   (add1₂₄)-prev value-(zero₂₃)
 end
 netFromPort (add₂₀)-return
@@ -70,10 +70,10 @@ netFromPort (mul₃)-return
   (add1₄₉)-prev value-(zero₃₉)
 end
 netFromPort (add1₅₇)-value!
-  (add1₅₇)-prev value-(add1₆₀)
-  (add1₆₀)-prev value-(add1₆₃)
-  (add1₆₃)-prev value-(add1₆₆)
-  (add1₆₆)-prev value-(zero₅₁)
+  (add1₅₇)-prev value-(add1₆₃)
+  (add1₆₃)-prev value-(add1₆₀)
+  (add1₆₀)-prev value-(add1₆₆)
+  (add1₆₆)-prev value-(zero₄₇)
 end
 netFromPort (mul₈)-return
   (mul₈)-target return-(add₄₇)
@@ -93,16 +93,16 @@ netFromPort (mul₈)-return
   (add1₆₈)-prev value-(zero₅₃)
   (add1₆₇)-prev value-(zero₅₂)
 end
-netFromPort (add1₇₉)-value!
-  (add1₇₉)-prev value-(add1₈₂)
-  (add1₈₂)-prev value-(add1₈₅)
-  (add1₈₅)-prev value-(add1₈₈)
+netFromPort (add1₇₈)-value!
+  (add1₇₈)-prev value-(add1₈₈)
   (add1₈₈)-prev value-(add1₉₇)
-  (add1₉₇)-prev value-(add1₉₁)
-  (add1₉₁)-prev value-(add1₉₄)
-  (add1₉₄)-prev value-(add1₁₀₀)
-  (add1₁₀₀)-prev value-(add1₁₀₃)
-  (add1₁₀₃)-prev value-(zero₇₀)
+  (add1₉₇)-prev value-(add1₈₁)
+  (add1₈₁)-prev value-(add1₈₄)
+  (add1₈₄)-prev value-(add1₉₁)
+  (add1₉₁)-prev value-(add1₁₀₀)
+  (add1₁₀₀)-prev value-(add1₉₄)
+  (add1₉₄)-prev value-(add1₁₀₃)
+  (add1₁₀₃)-prev value-(zero₆₄)
 end
 netFromPort (add1₁₀₇)-value!
   (add1₁₀₇)-prev value-(add1₁₀₄)
@@ -112,8 +112,8 @@ netFromPort (add1₁₁₂)-value!
   (add1₁₁₂)-prev value-(add1₁₁₃)
   (add1₁₁₃)-prev value-(zero₇₅)
 end
-netFromPort (add1₁₂₂)-value!
-  (add1₁₂₂)-prev value-(add1₁₂₃)
+netFromPort (add1₁₂₁)-value!
+  (add1₁₂₁)-prev value-(add1₁₂₃)
   (add1₁₂₃)-prev value-(add1₁₁₄)
   (add1₁₁₄)-prev value-(zero₇₇)
 end