This Sass code defines a recursive function to calculate Fibonacci numbers. The Fibonacci sequence is a fundamental concept in computer science and mathematics, often used to illustrate recursion. It's applied in areas l...

The `fibonacci` function calculates the nth Fibonacci number recursively. It has two base cases: when `n` is 0, it returns 0, and when `n` is 1, it returns 1. For any `n` greater than 1, it recursively calls itself with `n-1` and `n-2` and sums their results. This approach directly mirrors the mathematical definition of the Fibonacci sequence. The `safe_fibonacci` function acts as a wrapper, adding input validation to handle edge cases like negative numbers or non-integer inputs, preventing unexpected behavior or errors. The time complexity of this naive recursive approach is exponential, O(2^n), due to redundant calculations. The space complexity is O(n) due to the recursion depth on the call stack. While elegant, this implementation is inefficient for large `n`. A correctness argument can be made by induction: the base cases are correct, and if F(k) is correct for all k < n, then F(n) = F(n-1) + F(n-2) is also correct by definition.

Function fibonacci(n): If n is 0, return 0. If n is 1, return 1. Else, return fibonacci(n-1) + fibonacci(n-2). Function safe_fibonacci(n): If n is negative, throw error. If n is not an integer, throw error. Return fibona...

This Sass code implements an iterative binary search algorithm to find the index of a target element within a sorted array (represented as a Sass list). Binary search is a highly efficient search algorithm, commonly used...

The `binary_search` function takes a sorted Sass list and a target value, returning the index of the target if found, or -1 otherwise. It initializes `low` to 0 and `high` to the last index of the list. The `while` loop continues as long as `low` is less than or equal to `high`, indicating a valid search space. Inside the loop, the `mid` index is calculated, and the value at `mid` is compared to the `target`. If they match, `mid` is returned. If the `target` is greater than `mid_value`, the search continues in the right half by setting `low` to `mid + 1`. If the `target` is smaller, the search continues in the left half by setting `high` to `mid - 1`. An edge case for an empty array is handled at the beginning, returning -1 immediately. The time complexity of binary search is logarithmic, O(log n), because the search space is halved in each iteration. The space complexity is O(1) as it uses a constant amount of extra space for variables. The correctness is maintained by the invariant that the target, if present, always lies within the range [`low`, `high`].

Function binary_search(sorted_array, target): low = 0 high = length(sorted_array) - 1 If sorted_array is empty, return -1. While low <= high: mid = floor((low + high) / 2) mid_value = element at index mid in sorted_array...

This Sass mixin, `respond-to`, simplifies the process of applying responsive styles. It uses a map `$breakpoints` to store common screen widths associated with names like 'small', 'medium', and 'large'. When you include...

The `respond-to` mixin takes a `$breakpoint` name as an argument. It first checks if this name is a key in the `$breakpoints` map using `map-has-key`. If found, it retrieves the associated pixel value using `map-get` and then applies a `@media (min-width: $value)` query. The content within the mixin's block (`@content`) is then injected inside this media query. If the breakpoint name is not found, a warning is issued. This pattern is efficient, with a lookup time of O(1) on average for map operations and a direct application of the media query. It avoids repetitive media query syntax, making the Sass code more readable and less error-prone.

Define breakpoints map with names and pixel values. Mixin respond-to(breakpoint_name): If breakpoint_name exists in breakpoints map: Get pixel_value from breakpoints map for breakpoint_name. Apply media query min-width:...

This Sass mixin set, `generate-spacing-utilities`, along with its helpers `margin-utilities` and `padding-utilities`, creates a system of utility classes for applying consistent spacing (margins and paddings) throughout...

The `generate-spacing-utilities` mixin takes the CSS property ('margin' or 'padding'), a list of sides ('t', 'b', 'l', 'r', 'all'), and a map of spacing values. It iterates through each key-value pair in the `$values` map. For each spacing value, it then iterates through the specified `$sides`. It constructs a class name based on the property, side, and spacing key (e.g., `.m-t-1` for margin-top-1). If the side is 'all', it simplifies the class name (e.g., `.m-1`). Finally, it applies the corresponding CSS property and value to the generated class. The helper mixins `margin-utilities` and `padding-utilities` simply call the main mixin with predefined sides. The time complexity is O(S * V), where S is the number of sides and V is the number of spacing values, due to nested iteration. Space complexity is proportional to the number of generated utility classes.

Define spacing_values map (key: string, value: CSS unit). Mixin generate-spacing-utilities(property, sides, values): For each key, value in values: For each side in sides: If side is 'all': class_name = property + '-' +...

This Sass mixin, `grid-generator`, allows for the creation of a responsive grid system. It takes the desired number of columns and gutter width as primary arguments, along with an optional map of breakpoints. It generate...

The `grid-generator` mixin first validates the `$columns` and `$gutter` inputs. It then sets up base styles for `.grid-container` (using flexbox for layout) and `.grid-item` (handling box-sizing and padding). The core logic iterates through the provided `$breakpoints` map. For each breakpoint, it generates specific column classes (e.g., `.grid-item-X-breakpoint`) with calculated percentage widths. A loop (`@for`) is used to create classes for each possible column count up to the total number of columns for that breakpoint. It also ensures a default `.grid-item` width is set for the smallest breakpoint. The time complexity is roughly O(B * C), where B is the number of breakpoints and C is the number of columns, due to nested iteration and loops. Space complexity is proportional to the number of generated CSS classes.

Mixin grid-generator(columns, gutter, breakpoints): Validate columns and gutter inputs. Define base styles for .grid-container (display: flex, margins). Define base styles for .grid-item (box-sizing, padding). For each b...

This Sass mixin, `generate-theme-variables`, is designed to create a set of CSS custom properties (variables) for theming purposes. It takes a base color and an optional map of modifiers, applying adjustments like satura...

The `generate-theme-variables` mixin iterates through a map of color modifiers, applying Sass's `adjust-color` function to a provided base color. For each modifier, it generates a CSS custom property named `--theme-<modifier-name>` with the adjusted color value. The mixin includes a check to ensure the `$base-color` is indeed a color type, warning the user and exiting if it's not. It also ensures a `--theme-primary` variable is always set, using the `$base-color` if no 'primary' modifier is explicitly defined. This approach promotes a structured and maintainable way to manage color schemes in Sass projects, allowing for easy theme switching and consistent branding. The complexity is O(N) where N is the number of modifiers, as it iterates through them once. Space complexity is minimal, primarily for storing the generated variables.

Mixin generate-theme-variables(base_color, modifiers): If base_color is not a color: Warn user and return. For each name, modifier in modifiers: Calculate generated_color by adjusting base_color with modifier's saturatio...

This Sass mixin set, `generate-flex-alignment` and its helpers, creates a comprehensive library of utility classes for controlling flexbox alignment. It leverages a map `$flex-alignments` to store common flexbox alignmen...

The `generate-flex-alignment` mixin takes a CSS flexbox property (e.g., `justify-content`) and a map of alignment values. It iterates through the map, creating a class name by combining the property and the alignment key (e.g., `justify-content-center`). This class is then assigned the corresponding CSS value. The helper mixins (`justify-content-utilities`, `align-items-utilities`, `align-self-utilities`) call the main mixin with specific properties. The `flex-alignment-utilities` mixin combines all these helpers to generate a complete set of flexbox alignment utilities. The time complexity is O(P * A), where P is the number of properties being generated (e.g., justify-content, align-items, align-self) and A is the number of alignment values in the map. Space complexity is proportional to the number of generated CSS classes.

Define flex_alignments map (key: string, value: CSS flexbox value). Mixin generate-flex-alignment(property, alignments): For each key, value in alignments: class_name = property + '-' + key Define class .class_name with...

This Sass mixin, `fade-in`, provides a simple and reusable way to add a fade-in animation effect to HTML elements. It defines CSS keyframes for the animation and then applies them to the element using the `animation` sho...

The `fade-in` mixin defines a `@keyframes` rule named `fade-in`. This animation transitions an element's `opacity` from 0 (fully transparent) to 1 (fully opaque). The mixin then applies this animation to the element using the `animation-name`, `animation-duration`, `animation-delay`, `animation-timing-function`, and `animation-fill-mode` properties. `animation-fill-mode: forwards` is crucial as it ensures the element retains its final `opacity: 1` state after the animation completes. The mixin accepts parameters for duration, delay, and timing function, allowing for easy customization. The time complexity is O(1) as it generates a fixed set of CSS properties. Space complexity is also O(1) for the generated CSS rules.

Mixin fade-in(duration, delay, timing_function): Define keyframes fade-in: from: opacity: 0 to: opacity: 1 Apply animation: animation-name: fade-in animation-duration: duration animation-delay: delay animation-timing-fun...