You are an expert Verilog hardware designer working on a complete FPGA or ASIC project. You are given the full source code of a Verilog project. One of the modules in the project is incomplete — its implementation is missing and marked with:

// >>> Module Implementation Begin

// <<< Module Implementation End

Your task is to complete the missing implementation so that the entire project functions correctly. You must infer what the missing module or logic is supposed to do by analyzing the rest of the codebase: module names, signal connections, instantiations, comments, and usage patterns.

Follow these rules:

1. Only generate Verilog code to go between the `// >>> Module Implementation Begin` and `// <<< Module Implementation End` comments.
2. Use Verilog-2001 syntax. Ensure the code is synthesizable.
3. Assume the goal is a working, integrated hardware design — your code must make the whole system function as intended.
4. Analyze the naming, I/O ports, and how the module is instantiated or used in other parts of the project.
5. Implement correct behavior based on design context. Do not guess randomly — reason it out.
6. Avoid vendor-specific features unless explicitly required.
7. Do not output comments, explanation, or anything outside the required code region unless explicitly asked.

Your output should contain **only the Verilog code** that fits inside the marked region.

/*
 * Copyright (c) 2024 jeremy alcim
 * SPDX-License-Identifier: Apache-2.0
 */

`default_nettype none

module tt_um_top (
		       input wire [7:0]	 ui_in,	  // Dedicated inputs
		       output wire [7:0] uo_out,  // Dedicated outputs

		       input wire [7:0]	 uio_in,  // IOs: Input path
		       output wire [7:0] uio_out, // IOs: Output path
		       output wire [7:0] uio_oe, // IOs: Enable path (active high: 0=input, 1=output)

		       input wire ena,	// always 1 when the design is powered, so you can ignore it
		       input wire clk,	// clock
		       input wire rst_n	// reset_n - low to reset
		      );

   // All output pins must be assigned. If not used, assign to 0.
   assign uio_out = 0;
   assign uio_oe  = 0;

   // List all unused inputs to prevent warnings
   wire _unused = &{ena, uio_in[6],
		    i_oh_p[5], i_oh_p[4], i_oh_n[5], i_oh_n[4],
		    &{_ignore1}, &{_ignore2}, &{_ignore3}};

   /* verilator lint_off UNUSEDSIGNAL */
   wire [5:0] _ignore1;
   /* verilator lint_off UNUSEDSIGNAL */
   wire [5:0] _ignore2;
   /* verilator lint_off UNUSEDSIGNAL */
   wire [5:0] _ignore3;

////////////////////////////////////////////////////////////

   //input demux
   wire [5:0] i_demux_1, i_demux_2;
   assign i_demux_1 = ui_in [5:0];
   assign i_demux_2 = uio_in[5:0];

   //ctrl demux/mux
   wire [1:0] ctrl;
   assign ctrl = ui_in[7:6];

   demux #(.WAY(4), .WIRE(6)) input_one (i_demux_1, ctrl, demux_output_1);
   demux #(.WAY(4), .WIRE(6)) input_two (i_demux_2, ctrl, demux_output_2);

   //output demux
   wire [23:0] demux_output_1, demux_output_2;
   assign {cam_instr, _ignore1, i_oh_p, i_brent_A} = demux_output_1;
   assign {_ignore2, _ignore3, i_oh_n, i_brent_B} = demux_output_2;

////////////////////////////////////////////////////////////

   //input mux
   wire [31:0] i_mux;
   assign i_mux = {output_cam, output_vga, o_oh, o_brent};

   mux #(.WAY(4), .WIRE(8)) ouput_one (i_mux, ctrl, uo_out);

////////////////////////////////////////////////////////////

   //brent-kung
   
   // input 6 bit
   wire [5:0] i_brent_A, i_brent_B;

   // input 1 bit
   wire	      i_brent_Cin;
   assign i_brent_Cin = uio_in[7];

   brent_kung_cin brent(o_brent, i_brent_A, i_brent_B, i_brent_Cin);

   //brent-kung output
   wire [7:0] o_brent;

////////////////////////////////////////////////////////////

   //onehalf_latch
   
   wire [5:0] i_oh_p, i_oh_n;

   onehalf_latch onehalt_latch(clk, i_oh_p[0], i_oh_n[0], o_oh[1], o_oh[0]);
   onehalf_latch twohalt_latch(clk, i_oh_p[1], i_oh_n[1], o_oh[3], o_oh[2]);
   onehalf_latch treehalt_latch(clk, i_oh_p[2], i_oh_n[2], o_oh[5], o_oh[4]);
   onehalf_latch fourhalt_latch(clk, i_oh_p[3], i_oh_n[3], o_oh[7], o_oh[6]);

   wire	[7:0] o_oh;

////////////////////////////////////////////////////////////

   vga_example vga(output_vga, clk, rst_n);
   wire [7:0] output_vga;

////////////////////////////////////////////////////////////

   wire [5:0] cam_instr;
   assign {cam_write, cam_addr} = cam_instr;

   wire	      cam_write;
   wire [4:0] cam_addr;
   wire [7:0] cam_data;

   assign cam_data = uio_in;
   cam cam(output_addr, found, clk, ena, rst_n, cam_write, cam_addr, cam_data);
   wire [4:0] output_addr;
   wire	      found;

   wire [7:0] output_cam;
   assign output_cam = {2'b0, found, output_addr};

endmodule


module mux(input [SIZE_IN - 1:0] in,
	   input [SIZE_CTRL-1:0] ctrl,
	   output [WIRE-1:0]	 out);

   parameter WAY = 8;  // Nombre de voies par défaut
   parameter WIRE = 1; // taille de la sortie

   function integer log2;
      input integer value;
      begin
         value = value - 1;
         for (log2 = 0; value > 0; log2 = log2 + 1)
           value = value >> 1;
      end
   endfunction

   localparam SIZE_CTRL = log2(WAY);
   localparam SIZE_IN = WAY * WIRE;

   if (SIZE_CTRL == 1)
     /* verilator lint_off GENUNNAMED */
     assign out = ctrl ? in[2 * WIRE - 1 : WIRE] : in[WIRE - 1 : 0];
   else
     begin
	wire [WIRE-1:0]out1, out2;

	assign out = ctrl[SIZE_CTRL - 1] ? out2 : out1;
	mux #(.WAY(WAY/2), .WIRE(WIRE)) mux1(.in(in[(WAY/2) * WIRE - 1:0]),
					     .ctrl(ctrl[SIZE_CTRL - 2:0]),
					     .out(out1));

	mux #(.WAY(WAY/2), .WIRE(WIRE)) mux2(.in(in[SIZE_IN - 1 : (WAY/2) * WIRE]),
					     .ctrl(ctrl[SIZE_CTRL - 2:0]),
                                             .out(out2));

     end
endmodule

module demux(input [WIRE-1:0]	   in,
	     input [SIZE_CTRL-1:0] ctrl,
	     output [SIZE_OUT-1:0] out);

   parameter WAY = 8;   // Nombre de voies par défaut
   parameter WIRE = 1;  // Taille des sorties

   function integer log2;
      input integer value;
      begin
         value = value - 1;
         for (log2 = 0; value > 0; log2 = log2 + 1)
           value = value >> 1;
      end
   endfunction

   localparam SIZE_CTRL = log2(WAY);
   localparam SIZE_OUT  = WAY * WIRE;

   supply0 padding;

   if (SIZE_CTRL == 1)
     /* verilator lint_off GENUNNAMED */
     assign out = ctrl ? {in, {WIRE{padding}}} : {{WIRE{padding}}, in};
   else
     begin
        localparam N1 = (SIZE_OUT / 2) + (SIZE_OUT % 2);
        localparam N2 = SIZE_OUT / 2;

        wire [N1-1:0] W1;
        wire [N2-1:0] W2;

        assign out = ctrl[SIZE_CTRL-1] ? {W1, {N2{padding}}} : {{N1{padding}}, W2};
        demux #(.WAY(WAY/2), .WIRE(WIRE)) demux1(.in(in),
						 .ctrl(ctrl[SIZE_CTRL-2:0]),
						 .out(W1));

        demux #(.WAY(WAY/2), .WIRE(WIRE)) demux2(.in(in),
						 .ctrl(ctrl[SIZE_CTRL-2:0]),
						 .out(W2));
     end
endmodule


module black(P, G, Pi, Pj, Gi, Gj);
// >>> Module Implementation Begin

// <<< Module Implementation End
endmodule



module grey(P, G, A, B);
   output P,  G;
   input  A, B;

   assign P = A ^ B;
   assign G = A & B;

endmodule

module green(C_out, Pi, Gi, C_in);
  output C_out;
  input Pi, Gi, C_in;

  assign C_out = Gi | (Pi & C_in);

endmodule

module stage_0(P, G, A, B);
   output [3:0] P, G;
   input [3:0]	A, B;

   grey grey_0(P[0], G[0], A[0], B[0]);
   grey grey_1(P[1], G[1], A[1], B[1]);
   grey grey_2(P[2], G[2], A[2], B[2]);
   grey grey_3(P[3], G[3], A[3], B[3]);

endmodule

module brent_kung_cin (output_S, input_A, input_B, Cin);
   output [7:0] output_S;
   input	Cin;
   input [5:0]	input_A, input_B;

   wire [3:0]	A, B;

   /* verilator lint_off UNUSEDSIGNAL */
   wire [1:0]	_ignored = {input_A[5] & input_B[5], input_A[4] & input_B[4]};
   assign A = input_A[3:0];
   assign B = input_B[3:0];

   wire [4:0]	out;
   assign output_S = {3'b0, out};

////////////////////////////////////////////////////////////

   wire [2:0]	black_P, black_G;
   wire [4:0]	C;

   wire	[3:0]	GREY_P, GREY_G;
   stage_0 stage_0_0(GREY_P, GREY_G, B, A);
   assign C[0] = Cin;

   assign out[0] = GREY_P[0] ^ C[0];

   //stage 1
   green green_1(C[1], GREY_P[0], GREY_G[0], C[0]);//0
   assign out[1] = GREY_P[1] ^ C[1];

//   wire [3:0] stage_1_P, stage_1_G;
   black black_0(black_P[0], black_G[0], GREY_P[1], GREY_P[0], GREY_G[1], GREY_G[0]);//1
   black black_1(black_P[1], black_G[1], GREY_P[3], GREY_P[2], GREY_G[3], GREY_G[2]);//3

   //stage 2
   green green_2(C[2], black_P[0], black_G[0], C[0]);//1
   assign out[2] = GREY_P[2] ^ C[2];

//   wire [3:0] stage_2_P, stage_2_G;
   black black_4(black_P[2], black_G[2], black_P[1], black_P[0], black_G[1], black_G[0]);//3

   //stage 3
   green green_3(C[3], GREY_P[2], GREY_G[2], C[2]);//2
   assign out[3] = GREY_P[3] ^ C[3];
   green green_4(C[4], black_P[2], black_G[2], C[0]);//3
   assign out[4] = C[4];

endmodule
/*

 `default_nettype none

module tt_um_brent_kung (
		       input wire [7:0]	 ui_in,	  // Dedicated inputs
		       output wire [7:0] uo_out,  // Dedicated outputs

		       input wire [7:0]	 uio_in,  // IOs: Input path
		       output wire [7:0] uio_out, // IOs: Output path
		       output wire [7:0] uio_oe, // IOs: Enable path (active high: 0=input, 1=output)

		       input wire ena,	// always 1 when the design is powered, so you can ignore it
		       input wire clk,	// clock
		       input wire rst_n	// reset_n - low to reset
		      );

   // All output pins must be assigned. If not used, assign to 0.
   //  assign uo_out  = ui_in + uio_in;  // Example: ou_out is the sum of ui_in and uio_in
   brent_kung_cin brent(uo_out[4:0], ui_in[3:0], uio_in[3:0], 1'b0);
   assign uio_out = 0;
   assign uio_oe  = 0;

   // List all unused inputs to prevent warnings
   wire _unused = &{ena, clk, rst_n, 1'b0, ui_in[7:4], uio_in[7:4]};
   assign uo_out[7:5] = 3'b0;

endmodule
*/


/* 
  Public Domain
  Lionel Sainte Cluque
  2025/10/14 rev alpha


This file implements the latch for a 1.5bit Digital to analog sigma delta modulator
It takes to input (from comparators) and output a 2 bit wide output to drive power stage.
As the two transistors can not be on at the same time the output can take three values only : 
  * 00
  * 01 
  * 10
But never 11


Reference : 
Design of a 3rd Order 1.5-Bit Continuous-Time Fully
Differential Sigma-Delta (Σ∆) Modulator Optimized for
a Class D Audio Amplifier Using Differential Pairs
Nuno Pereira, João Melo, Nuno Paulino
*/

module onehalf_latch(input  clk,
		     input in_p,
		     input in_n,

		     output out_p,
		     output out_n);

   reg reg_p, reg_n;
   wire	forbidden;

   always @(posedge clk)
     begin 
	reg_p <= in_p;
	reg_n <= in_n;
     end

   assign forbidden = ~(reg_p & reg_n);
   assign out_p = reg_p & forbidden;
   assign out_n = reg_n & forbidden;

endmodule


/*
 * Copyright (c) 2024 Uri Shaked
 * SPDX-License-Identifier: Apache-2.0
 */

`default_nettype none

module vga_example(
  output wire [7:0] uo_out,   // Dedicated outputs
  input  wire       clk,      // clock
  input  wire       rst_n     // reset_n - low to reset
);
  // VGA signals
  wire hsync;
  wire vsync;
  wire [1:0] R;
  wire [1:0] G;
  wire [1:0] B;
  wire video_active;
  wire [9:0] pix_x;
  wire [9:0] pix_y;

  // TinyVGA PMOD
  assign uo_out = {hsync, B[0], G[0], R[0], vsync, B[1], G[1], R[1]};

   /* verilator lint_off UNUSEDSIGNAL */
  wire _ignore = |{pix_y[9], pix_y[8], pix_y[7], pix_y[6], pix_y[4], pix_y[3], pix_y[1], pix_y[0],
		   moving_x[9],moving_x[8],moving_x[4],moving_x[3],moving_x[2],moving_x[1],moving_x[0]};

  reg [9:0] counter;

  hvsync_generator hvsync_gen(
    .clk(clk),
    .reset(~rst_n),
    .hsync(hsync),
    .vsync(vsync),
    .display_on(video_active),
    .hpos(pix_x),
    .vpos(pix_y)
  );
  
  wire [9:0] moving_x = pix_x + counter;

  assign R = video_active ? {moving_x[5], pix_y[2]} : 2'b00;
  assign G = video_active ? {moving_x[6], pix_y[2]} : 2'b00;
  assign B = video_active ? {moving_x[7], pix_y[5]} : 2'b00;

  always @(posedge vsync, negedge rst_n) begin
    if (~rst_n) begin
      counter <= 0;
    end else begin
      counter <= counter + 1;
    end
  end
  
endmodule // tt_um_vga_example

`ifndef HVSYNC_GENERATOR_H
`define HVSYNC_GENERATOR_H

/*
Video sync generator, used to drive a VGA monitor.
Timing from: https://en.wikipedia.org/wiki/Video_Graphics_Array
To use:
- Wire the hsync and vsync signals to top level outputs
- Add a 3-bit (or more) "rgb" output to the top level
*/

module hvsync_generator(clk, reset, hsync, vsync, display_on, hpos, vpos);

  input clk;
  input reset;
  output reg hsync, vsync;
  output display_on;
  output reg [9:0] hpos;
  output reg [9:0] vpos;

  // declarations for TV-simulator sync parameters
  // horizontal constants
  parameter H_DISPLAY       = 640; // horizontal display width
  parameter H_BACK          =  48; // horizontal left border (back porch)
  parameter H_FRONT         =  16; // horizontal right border (front porch)
  parameter H_SYNC          =  96; // horizontal sync width
  // vertical constants
  parameter V_DISPLAY       = 480; // vertical display height
  parameter V_TOP           =  33; // vertical top border
  parameter V_BOTTOM        =  10; // vertical bottom border
  parameter V_SYNC          =   2; // vertical sync # lines
  // derived constants
  parameter H_SYNC_START    = H_DISPLAY + H_FRONT;
  parameter H_SYNC_END      = H_DISPLAY + H_FRONT + H_SYNC - 1;
  parameter H_MAX           = H_DISPLAY + H_BACK + H_FRONT + H_SYNC - 1;
  parameter V_SYNC_START    = V_DISPLAY + V_BOTTOM;
  parameter V_SYNC_END      = V_DISPLAY + V_BOTTOM + V_SYNC - 1;
  parameter V_MAX           = V_DISPLAY + V_TOP + V_BOTTOM + V_SYNC - 1;

  wire hmaxxed = (hpos == H_MAX) || reset;	// set when hpos is maximum
  wire vmaxxed = (vpos == V_MAX) || reset;	// set when vpos is maximum
  
  // horizontal position counter
  always @(posedge clk)
  begin
    hsync <= (hpos>=H_SYNC_START && hpos<=H_SYNC_END);
    if(hmaxxed)
      hpos <= 0;
    else
      hpos <= hpos + 1;
  end

  // vertical position counter
  always @(posedge clk)
  begin
    vsync <= (vpos>=V_SYNC_START && vpos<=V_SYNC_END);
    if(hmaxxed)
      if (vmaxxed)
        vpos <= 0;
      else
        vpos <= vpos + 1;
  end
  
  // display_on is set when beam is in "safe" visible frame
  assign display_on = (hpos<H_DISPLAY) && (vpos<V_DISPLAY);

endmodule

`endif


module cam(output [4:0]	out,
	   output reg	found,

	   input	clk,
	   input	enable,
	   input	rst_n,
	   input	write,
	   input [4:0]	addr,
	   input [7:0]	data);

   parameter NB_MEM = 16;
   parameter SIZE_ADDR = 4;

   reg [7:0] mem [0:NB_MEM-1];
   wire [SIZE_ADDR-1:0] ret;

   /* verilator lint_off UNUSEDSIGNAL */
   wire	   _ignore = addr[4];

   wire [NB_MEM-1:0] match;
   wire [SIZE_ADDR-1:0] addr_out [0:NB_MEM-1];

   genvar i;
   generate
      for (i = 0; i < NB_MEM; i = i + 1)
	begin : gen_match
	   assign match[i] = (mem[i] == data);
	   assign addr_out[i] = match[i] ? i[SIZE_ADDR-1:0] : {SIZE_ADDR{1'b0}};
	end
   endgenerate

   assign ret = write ? 4'b0 :
		addr_out[0] | addr_out[1] | addr_out[2] | addr_out[3] |
		addr_out[4] | addr_out[5] | addr_out[6] | addr_out[7] |
		addr_out[8] | addr_out[9] | addr_out[10] | addr_out[11] |
		addr_out[12] | addr_out[13] | addr_out[14] | addr_out[15];

   always @(posedge clk or negedge rst_n)
     begin

	if (~rst_n)
	  begin
	     found <= 0;
	     /*
	     mem[0] <= 8'h00;
	     mem[1] <= 8'h00;
	     mem[2] <= 8'h00;
	     mem[3] <= 8'h00;
	     mem[4] <= 8'h00;
	     mem[5] <= 8'h00;
	     mem[6] <= 8'h00;
	     mem[7] <= 8'h00;
	     mem[8] <= 8'h00;
	     mem[9] <= 8'h00;
	     mem[10] <= 8'h00;
	     mem[11] <= 8'h00;
	     mem[12] <= 8'h00;
	     mem[13] <= 8'h00;
	     mem[14] <= 8'h00;
	     mem[15] <= 8'h00;
	      */
	  end

	else if (write & enable)
	  mem[addr[SIZE_ADDR-1:0]] <= data;

	else if (enable)
	  begin
	     found <= |match;
	  end

     end

   assign out = {1'b0, ret};
endmodule


Reference solution:
rtl/generated.v:    output P,  G;
   input  Pi, Pj, Gi, Gj;

   assign P = Pi & Pj;
   assign G = Gi | (Pi & Gj);